Recent zbMATH articles in MSC 20https://zbmath.org/atom/cc/202022-09-13T20:28:31.338867ZUnknown authorWerkzeugA structure-oriented construction of the classical number domains. With a view to order structures, algebraic and topological structureshttps://zbmath.org/1491.000022022-09-13T20:28:31.338867Z"Maurer, Christian"https://zbmath.org/authors/?q=ai:maurer.christianPublisher's description: Dieses Buch entwickelt systematisch die Konstruktion der klassischen Zahlenbereiche mit Blick auf die wichtigsten mathematischen Strukturen: Ordnungsstrukturen, algebraische Strukturen und topologische Strukturen. Kurze Zusammenfassungen je Kapitel/Abschnitt erleichtern die Übersicht und das Verinnerlichen der Inhalte. Das Buch bietet einen weit vernetzten Überblick über die fachwissenschaftlichen Grundlagen und deren zentrale Zusammenhänge; damit dient es Studierenden im Fach- und insbesondere im Lehramtsstudium Mathematik für Grundschule und Sekundarstufe I als wertvolle Ergänzung und Begleitung während der ersten Semester. Lehrende finden hier eine Alternative zum klassischen Einstieg ins Studium. Darüber hinaus ist das Buch auch für Quereinsteiger -- etwa Lehrkräfte anderer Fächer -- zur berufsbegleitenden Weiterbildung geeignet.On first order rigidity for linear groupshttps://zbmath.org/1491.030292022-09-13T20:28:31.338867Z"Plotkin, Eugene"https://zbmath.org/authors/?q=ai:plotkin.eugene|plotkin.eugene-b|plotkin.eugene-iSummary: The paper is a short survey of recent developments in the area of first order descriptions of linear groups. It is aimed to illuminate the known results and to pose the new problems relevant to logical characterizations of Chevalley groups and Kac-Moody groups.Rejewski: Permutation theory for cracking Enigmahttps://zbmath.org/1491.050082022-09-13T20:28:31.338867Z"Jiménez Seral, Paz"https://zbmath.org/authors/?q=ai:jimenez-seral.paz(no abstract)Bijections between compositions over finite groupshttps://zbmath.org/1491.050162022-09-13T20:28:31.338867Z"Gao, Zhicheng"https://zbmath.org/authors/?q=ai:gao.zhicheng"Zhang, Tiancheng"https://zbmath.org/authors/?q=ai:zhang.tianchengSummary: One may generalize integer compositions by replacing positive integers with elements from an additive group, giving the broader concept of compositions over a group. In this note we give some simple bijections between compositions over a finite group. It follows from these bijections that the number of compositions of a nonzero group element \(g\) is independent of \(g\). As a result we derive a simple expression for the number of compositions of any given group element. This extends an earlier result for abelian groups which was obtained using generating functions and a multivariate multisection formula.On the spectrum of Cayley graphshttps://zbmath.org/1491.051012022-09-13T20:28:31.338867Z"Ghorbani, M."https://zbmath.org/authors/?q=ai:ghorbani.modjtaba"Songhori, M."https://zbmath.org/authors/?q=ai:songhori.mahinSummary: The set of eigenvalues of the adjacency matrix of a graph is called the spectrum of it. In the present paper, we introduce the spectrum of Cayley graphs of order \(pqr\) in terms of character table, where \(p, q, r\) are prime numbers. We also, stablish some properties of Cayley graphs of non-abelian groups with a normal symmetric connected subset.The poset of morphism-extension classes of countable graphshttps://zbmath.org/1491.051362022-09-13T20:28:31.338867Z"Aranda, Andrés"https://zbmath.org/authors/?q=ai:aranda.andresSummary: Let \(\mathrm{XY}_{L,T}\) consist of all countable \(L\)-structures \(M\) that satisfy the axioms \(T\) and in which all homomorphisms of type X (these could be plain homomorphisms, monomorphisms, or isomorphisms) between finite substructures of \(M\) are restrictions of an endomorphism of \(M\) of type Y (for example, an automorphism or a surjective endomorphism). Lockett and Truss introduced 18 such classes for relational structures. For a given pair \(L, T\) however, two or more morphism-extension properties may define the same class of structures.
In this paper, we establish all equalities and inequalities between morphism-extension classes of countable graphs.The classification of multiplicity-free plethysms of Schur functionshttps://zbmath.org/1491.051882022-09-13T20:28:31.338867Z"Bessenrodt, Christine"https://zbmath.org/authors/?q=ai:bessenrodt.christine"Bowman, Chris"https://zbmath.org/authors/?q=ai:bowman.christopher-david"Paget, Rowena"https://zbmath.org/authors/?q=ai:paget.rowenaSummary: We classify and construct all multiplicity-free plethystic products of Schur functions. We also compute many new (infinite) families of plethysm coefficients, with particular emphasis on those near maximal in the dominance ordering and those of small Durfee size.Skew shape asymptotics, a case-based introductionhttps://zbmath.org/1491.051942022-09-13T20:28:31.338867Z"Pak, Igor"https://zbmath.org/authors/?q=ai:pak.igorSummary: We discuss various tools in the emerging area of Asymptotic Algebraic Combinatorics, as they apply to one running example of thick ribbons. Connections to other areas, exercises and open problems are also included.Uncrowding algorithm for hook-valued tableauxhttps://zbmath.org/1491.051952022-09-13T20:28:31.338867Z"Pan, Jianping"https://zbmath.org/authors/?q=ai:pan.jianping"Pappe, Joseph"https://zbmath.org/authors/?q=ai:pappe.joseph"Poh, Wencin"https://zbmath.org/authors/?q=ai:poh.wencin"Schilling, Anne"https://zbmath.org/authors/?q=ai:schilling.anneA hook tableau is a semi-standard Young tableau shaped like an `L', in French notation. A hook-valued tableau is a tableau where each box contains a hook tableau, such that
\par i) if a box \(A\) is to the left of a box \(B\), but in the same row, then \(\max(A) \leqslant \min(B)\), and
\par ii) if a box \(A\) is below a box \(B\), but in the same column, then \(\max(A) < \min(C)\).
Here, \(\max(A)\) refers to the maximal entry of the hook tableau in box \(A\), and \(\min(A)\) is the minimal entry. Hook-valued tableaux generalise set-valued tableau and multiset-valued tableau: these result from the cases where the hooks consist of single columns and single rows, respectively. Just as with other sorts of tableaux, there is a crystal structure on hook-valued tableaux, introduced by \textit{G. Hawkes} and \textit{T. Scrimshaw} [Algebr. Comb. 3, No. 3, 727--755 (2020; Zbl 1441.05236)]. This specialises in the crystal structure of set-valued tableaux and multiset-valued tableaux.
For set-valued tableaux, there exists an uncrowding operator which maps a set-valued tableau to a pair consisting of a semi-standard Young tableau and a flagged increasing tableau. The operator is ``uncrowding'' in the sense that the output semistandard Young tableau has the same underlying multiset of numerical entries as the original set-valued tableau, except now we have one numerical entry per box, instead of a set of numerical entries per box. The flagged increasing tableau which is part of the output records data on how the tableau was uncrowded, thus allowing the original tableau to be reconstructed from the pair. A flagged increasing tableau is a tableau of skew shape which is increasing in both rows and columns such that entries in row \(i\) are at least \(i - 1\). An important property of the uncrowding operator on set-valued tableaux is that it intertwines with crystal operators.
The heart of the paper is the definition of an uncrowding operator for hook-valued tableaux. The output of such an operator is a set-valued tableaux and a column-flagged increasing tableaux. A column-flagged increasing tableau is the transpose of a flagged increasing tableau. This operator also has the desired property of intertwining with crystal operators. One can then uncrowded completely by uncrowding the output set-valued tableau.
There also exists an uncrowding operator on multiset-valued tableaux. The authors prove that their uncrowding operator on hook-valued tableaux generalises this operator. They also provide an inverse to their uncrowding map, giving a ``crowding'' map, which reassembles the original hook-valued tableau from a set-valued tableau and a column-flagged increasing tableau. This crowding map can only be applied to pairs that are compatible with each other in a certain way.
The authors also introduce an alternative uncrowding map on hook-valued tableaux which outputs a multiset-valued tableau and flagged increasing tableau. This uncrowding operator uncrowds the legs of the hooks in the hook-valued tableau, rather than the arms, as it were. It likewise intertwines with crystal operators.
In the final section, the authors apply their results to canonical Grothendieck polynomials. They use the uncrowding map to show that canonical Grothendieck polynomials have a tableau Schur expansion. Canonical Grothendieck polynomials are symmetric polynomials that can be expressed as generating functions of hook-valued tableaux. A symmetric function is said to have a tableau Schur expansion if it is the weighted sum of the Schur functions of a particular set of tableaux. A corollary of this result is an expansion of canonical Grothendieck polynomials in terms of stable symmetric Grothendieck polynomials and dual stable symmetric Grothendieck polynomials. Here, stable symmetric Grothendieck polynomials are generating functions of set-valued tableau and dual stable symmetric Grothendieck polynomials are generating functions of reverse plane partitions.
Reviewer: Nicholas Williams (Cologne)Cubic graphical regular representations of \(\mathrm{PSU}_3(q)\)https://zbmath.org/1491.051992022-09-13T20:28:31.338867Z"Li, Jing Jian"https://zbmath.org/authors/?q=ai:li.jingjian"Xia, Binzhou"https://zbmath.org/authors/?q=ai:xia.binzhou"Zhang, Xiao Qian"https://zbmath.org/authors/?q=ai:zhang.xiaoqian"Zheng, Shasha"https://zbmath.org/authors/?q=ai:zheng.shashaSummary: A graphical regular representation (GRR) of a group \(G\) is a Cayley graph of \(G\) whose full automorphism group is equal to the right regular permutation representation of \(G\). Towards a proof of the conjecture that only finitely many finite simple groups have no cubic GRR, this paper shows that \(\mathrm{PSU}_3(q)\) has a cubic GRR if and only if \(q \geq 4\). Indeed, for each \(q\), we construct explicitly a cubic GRR of \(\mathrm{PSU}_3(q)\).Semigroups in complete lattices. Quantales, modules and related topicshttps://zbmath.org/1491.060012022-09-13T20:28:31.338867Z"Eklund, Patrik"https://zbmath.org/authors/?q=ai:eklund.patrik-e"Gutiérrez García, Javier"https://zbmath.org/authors/?q=ai:gutierrez-garcia.javier"Höhle, Ulrich"https://zbmath.org/authors/?q=ai:hohle.ulrich"Kortelainen, Jari"https://zbmath.org/authors/?q=ai:kortelainen.jariThis monograph is a detailed and extensive investigation of the theory of quantales in different areas of mathematics. The unifying element is the adequate and skillful use of category theory. In mathematics, quantales are certain partially ordered algebraic structures that generalize locales (point free topologies) as well as various multiplicative lattices of ideals from ring theory and functional analysis (C*-algebras, von Neumann algebras). A quantale is a complete lattice \(Q\) with an associative binary operation \(*\) such that \(a*-\) and \(-*a\) both preserve arbitrary supremums for any \(a\in Q\). The term \textit{quantale} was introduced by \textit{C. J. Mulvey} [Rend. Circ. Mat. Palermo (2) Suppl. No. 12, 99--104 (1986; Zbl 0633.46065)] as a combination of ``quantum logic'' and ``locale''. Quantales are sometimes referred to as complete residuated semigroups.
It seems that there are two approaches to the theory of quantales. The first approach is residuation. W. Krull was first who formulated the axioms of ideal lattices as those axioms of a two-sided quantale. He also formulated the axioms of prime elements of two-sided quantales and noticed the existence of left residuals and right residuals. The first simple properties of residuals had already been collected in a paper (see the ref. [69]), although some formulas derived by R. Dedekind for modules in the commutative setting. The book by \textit{K. I. Rosenthal} [Quantales and their applications. Harlow: Longman Scientific \& Technical; New York: John Wiley \& Sons, Inc. (1990; Zbl 0703.06007)] concern with this approach and tends to concentrate on the algebraic rather than the topological aspects of quantale theory.
The existence of the tensor product together with the important fact that the category \textbf{Sup} of complete lattices and join-preserving maps is symmetric and monoidal closed opens a completely different view on the theory of quantales than the previous perspective from the point of view of residuation. First note that the category \textbf{Sup} is star-autonomous and also it is the Eilenberg-Moore category of the monad of down sets on the category of preordered sets. Second, quantales are indeed semigroups in \textbf{Sup}, and the formation of right- (left-)residuals determines a right (left) action on the dual lattice of the underlying quantale. Hence, the most natural applications of quantales arise in the theory of modules on unital quantale. This second approach to the theory of quantales was initiated by \textit{A. Joyal} and \textit{M. Tierney} [An extension of the Galois theory of Grothendieck. Providence, RI: American Mathematical Society (AMS) (1984; Zbl 0541.18002)]. Here, they observed the interesting fact that frames are special commutative monoids in \textbf{Sup}. The existence of free quantales is an immediate corollary of the general theorem that in every cocomplete, symmetric, and monoidal closed category free semigroups exist. Based on this observation, the cocompleteness of the category of semigroups in \textbf{Sup} is inherited by \textbf{Sup}. From this point of view the authors take categorical aspects of preordered sets as a natural starting point for a more fundamental understanding of quantales and their related topics.
One can essentially divide the book under review into four parts: (i) Categorical aspects of preordered sets; completeness and cocompleteness, cartesian closedness, free preordered groupoids, the term monad (of a single binary operator symbol) and the monad of down sets, (ii) Monoid theory in \textbf{Sup}, (iii) Module theory in \textbf{Sup} and (iv) an Appendix which contains all materials needed to follow the text.
In more detail, the organization of the book is as follows:
In Chapter 1, entitled ``Foundations'', the authors give definitions, notations, and basic results needed to understand the rest of the book. Also, the necessary categorical tools are introduced. In particular, the preliminaries are kept modest by including a detailed account of monoidal categories, magmas, semigroups, and monoids therein, free constructions, and tensor products. This chapter is divided into three sections: Some Properties of Universal Algebra in Monoidal Categories, Monads and Distributive Laws, The Category of Preordered Sets.
Chapter 2 is devoted to study a survey of the most important properties of the theory of (pre)quantales from the perspective of the category \textbf{Sup}. In the first section, a detailed account of the categorical properties of \textbf{Sup} is presented. Also the importance of closure operators for quotient constructions in \textbf{Sup} is explained. The last construction that mentioned here is the construction of tensor product in \textbf{Sup} which turns \textbf{Sup} into a symmetric monoidal closed category. In section 2, prequantals are introduced. A prequantale is a complete lattice \(X\) with a binary operation \(*: X\times X\to X\) which is a bimorphism of \textbf{Sup} i.e., \(*\) is a join-preserving map in each components (it is a quantale if the multiplication \(*\) is associative). It is proved that the category \textbf{Pq} of prequantales is isomorphic to the Eilenberg-Moore category of the composite monad \(\mathbb{D}wn\circ \mathbb{T}\) on \textbf{Preord} where \(\mathbb{D}wn\) and \(\mathbb{T}\) are the monad of down sets and the term monad on the category \textbf{Preord} of preordered sets, respectively. The structure of free prequantales is discussed. Then they study involutive prequantales and show that the categories \textbf{Pq} of prequantales and \textbf{IPq} of involutive prequantales are algebraic. (A involutive prequantale is a prequantale \((X, *)\) together with an isotone map \(\iota_X : X\to X\) such that \(\iota_X \circ\iota_X = id_X\).) This chapter is divided into seven sections. The other subjects disscussed here are: Quantales, Idempotent Quantales and Frames, Balanced and Bisymmetric Quantales, Frobenius Quantales, Complete MV-Algebras. A triple \((X, *, \delta )\) is called a Frobenius quantale if \((X, *)\) is a quantale, \(d\in X\) and \(\delta : 1\to X^{op}\) is the join-preserving map with \(\delta (1) = d\) (for more detail see Def. 2.6.1 on page 166). Also, a Frobenius quantale \((X, *, \delta )\) is called a Girard quantale if the dualizing element \(d\) is cyclic. In section 7, the authors consider the construction of Girard quantales from the perspective of residuated, commutative lattices. To do so, they define a Girard algebra as a residuated preordered groupoid \((X, *, \searrow \ , \ \swarrow )\), which satisfies the additional properties (i) the underlying preordered set is a bounded lattice, (ii) \((X, *)\) is a commutative semigroup and (iii) the universal lower bound is a dualizing element. Here, \(\searrow\) and \(\swarrow\) denote the left and right residuals. Since the left- and right- residuals coincide, we simply write \((X, *, \rightarrow )\) instead of \((X, *, \searrow \ , \ \swarrow )\). An MV-algebra is a Girard algebra in which for all \(x, y\in X\) the relation \((x \rightarrow y) \rightarrow y = x \vee y\) holds (MV-Property). The last result of this section states that every infinite, simple and complete MV-algebra is isomorphic to the real unit interval provided with the Lukasiewicz arithmetic conjunction (see Theorem 2.7.18 on page 195).
The third chapter of the monograph is about applications of quantales to module theory. In practice, to compare this approach with the traditional theory of modules over rings, this means that we can replace the category of abelian groups by the category of complete lattices and join-preserving maps. As a non-trivial applications to the theory of operator algebras, they show that irreducible representations of a \(C^*\)-algebra \(A\) and irreducible involutive left modules on the unital quantale of all closed linear subspaces of \(A\) are equivalent concepts. As another application they provide the fact that the category of right modules on a unital quantale plays the same role as \textbf{Sup} has done in the binary setting given by the two-element chain \(\{ 0 , 1\}\). Furthermore, if we restrict our attention to commutative and unital quantales \(Q\) then the category of modules over \(Q\) has a tensor product which is a quotient of the tensor product in \textbf{Sup}. This implies that the category of modules over \(Q\) is star-autonomous and we can repeat all those algebraic constructions in \textbf{Sup} in this new setting. The chapter finishes with a small section on automata in Sup showing that every automaton gives rise to a right module over the free unital quantale generated by its input alphabet. These applications illustrate the significance of quantals for further research across mathematics and computer science with an interest in categoty theory, lattice theory, and maney-valued logic.
Basically, the authors have taken the indicated material, organized it efficiently, written a very lucid, readable development of it in 316 pages, and added helpful historical remarks and a few exercises to each chapter and a brief appendix on the construction of free functor algebras and the role of coherence axioms in the construction of the tensor product of semigroups in symmetric monoidal categories. In conclusion the book is self-contained, well-organized and well-written and is warmley recommended to read.
Reviewer: Ali Madanshekaf (Semnan)Complemented lattices of subrackshttps://zbmath.org/1491.060162022-09-13T20:28:31.338867Z"Saki, A."https://zbmath.org/authors/?q=ai:saki.a"Kiani, D."https://zbmath.org/authors/?q=ai:kiani.dariushSummary: In this paper, a question due to \textit{I. Heckenberger} et al. [Trans. Am. Math. Soc. 372, No. 2, 1407--1427 (2019; Zbl 07076699)] on racks is positively answered. A rack is a set together with a self-distributive bijective binary operation. We show that the lattice of subracks of every finite rack is complemented. Moreover, we characterize finite modular lattices of subracks in terms of complements of subracks. Also, we introduce a certain class of racks including all finite groups with the conjugation operation, called \(G\)-racks, and we study some of their properties. In particular, we show that a finite \(G\)-rack has the homotopy type of a sphere. Further, we show that the lattice of subracks of an infinite rack is not necessarily complemented which gives an affirmative answer to the aforementioned question. Indeed, we show that the lattice of subracks of the set of rational numbers, as a dihedral rack, is not complemented. Finally, we show that being a Boolean algebra, pseudocomplemented and uniquely complemented as well as distributivity are equivalent for the lattice of subracks of a rack.A relationship between the category of chain MV-algebras and a subcategory of abelian groupshttps://zbmath.org/1491.060312022-09-13T20:28:31.338867Z"Pajoohesh, Homeira"https://zbmath.org/authors/?q=ai:pajoohesh.homeiraSummary: The category of MV-algebras is equivalent to the category of abelian lattice ordered groups with strong units. In this article we introduce the category of circled abelian groups and prove that the category of chain MV-algebras is isomorphic with the category of chain circled abelian groups. In the last section we show that the category of chain MV-algebras is a subcategory of abelian cyclically ordered groups.From \(\vee e\)-semigroups to hypersemigroupshttps://zbmath.org/1491.060392022-09-13T20:28:31.338867Z"Kehayopulu, Niovi"https://zbmath.org/authors/?q=ai:kehayopulu.nioviSummary: A \(poe\)-semigroup is a semigroup \(S\) at the same time an ordered set having a greatest element ``\(e\)'' in which the multiplication is compatible with the ordering. A \(\vee e\)-semigroup is a semigroup \(S\) at the same time an upper semilattice with a greatest element ``\(e\)'' such that \(a(b\vee c)= ab\vee ac\) and \((a\vee b)c= ac\vee bc\) for every \(a,b,c\in S\). If \(S\) is not only an upper semilattice but a lattice, then it is called \(\vee e\)-semigroup. From many results on \(\vee e\)-semigroups, \(\vee e\)-semigroups or \(poe\)-semigroups, corresponding results on ordered semigroups (without greatest element) can be obtained. Related results on hypersemigroups or ordered hypersemigroups follow as application. An example is presented in the present note; the same can be said for every result on these structures. So order-lattices play an essential role in studying the hypersemigroups and the ordered hypersemigroups.On automorphisms of monotone transformation posemigroupshttps://zbmath.org/1491.060402022-09-13T20:28:31.338867Z"Mir, Dilawar Juneed"https://zbmath.org/authors/?q=ai:mir.dilawar-juneed"Shah, Aftab Hussain"https://zbmath.org/authors/?q=ai:shah.aftab-hussain"Ahanger, Shabir Ahmad"https://zbmath.org/authors/?q=ai:ahanger.shabir-ahmadOn a generalization of principal weak (po-)flatness of \(S\)-posetshttps://zbmath.org/1491.060422022-09-13T20:28:31.338867Z"Rashidi, Hamideh"https://zbmath.org/authors/?q=ai:rashidi.hamideh"Golchin, Akbar"https://zbmath.org/authors/?q=ai:golchin.akbar"Saany, Hossein Mohammadzadeh"https://zbmath.org/authors/?q=ai:saany.hossein-mohammadzadeh\(S\)-acts over a well-ordered monoid with modular congruence latticehttps://zbmath.org/1491.060432022-09-13T20:28:31.338867Z"Stepanova, Alena A."https://zbmath.org/authors/?q=ai:stepanova.alena-andreevnaSummary: This work relates to the structural act theory. The structural theory includes the description of acts over certain classes of monoids or having certain properties, for example, satisfying some requirement for the congruence lattice. The congruences of universal algebra is the same as the kernels of homomorphisms from this algebra into other algebras. Knowledge of all congruences implies the knowledge of all the homomorphic images of the algebra. A left \(S\)-act over monoid \(S\) is a set \(A\) upon which \(S\) acts unitarily on the left. In this paper, we consider \(S\)-acts over linearly ordered and over well-ordered monoids, where a linearly ordered monoid \(S\) is a linearly ordered set with a minimal element and with a binary operation \(max\), with respect to which \(S\) is obviously a commutative monoid; a well-ordered monoid \(S\) is a well-ordered set with a binary operation \(max\), with respect to which \(S\) is also a commutative monoid. The paper is a continuation of the work of the author in co-authorship with \textit{M. S. Kazak} [Sib. Èlektron. Mat. Izv. 16, 1147--1157 (2019; Zbl 1472.06019)], which describes \(S\)-acts over linearly ordered monoids with a linearly ordered congruence lattice and \(S\)-acts over a well-ordered monoid with distributive congruence lattice. In this article, we give the description of S-acts over a well-ordered monoid such that the corresponding congruence lattice is modular.Some generalizations of p-semisimple BCI algebras and groupshttps://zbmath.org/1491.060582022-09-13T20:28:31.338867Z"Walendziak, Andrzej"https://zbmath.org/authors/?q=ai:walendziak.andrzejSummary: We introduce and investigate the strong p-semisimple property for some generalizations of BCI algebras. For BCI algebras, the strong p-semisimple property is equivalent to the p-semisimple property. We describe the connections of strongly p-semisimple algebras and various generalizations of groups (such as, for example, involutive moons and goops). Moreover, we present some examples of proper strongly p-semisimple algebras.A study of generalized quasi-hyperideals in ordered ternary semihypergroupshttps://zbmath.org/1491.060592022-09-13T20:28:31.338867Z"Abbasi, Mohammad Y."https://zbmath.org/authors/?q=ai:abbasi.mohammad-yahya"Hila, Kostaq"https://zbmath.org/authors/?q=ai:hila.kostaq"Khan, Sabahat A."https://zbmath.org/authors/?q=ai:khan.sabahat-ali"Talee, Aakif F."https://zbmath.org/authors/?q=ai:talee.aakif-fSummary: In this paper, we introduce and study generalized quasi-hyperideals in ordered ternary semihypergroups. Also, we define some generalized kinds of hyperideals in ordered ternary semihypergroups and study the relation between them.On ordered bi-\(\Gamma\)-ideals in ordered \(\Gamma\)-semigroupshttps://zbmath.org/1491.060602022-09-13T20:28:31.338867Z"Basar, Abul"https://zbmath.org/authors/?q=ai:basar.abul"Abbasi, Mohammad Yahya"https://zbmath.org/authors/?q=ai:abbasi.mohammad-yahya(no abstract)On 0-minimal (0,2)-bi-hyperideal of ordered semihypergroupshttps://zbmath.org/1491.060612022-09-13T20:28:31.338867Z"Hobanthad, Samkhan"https://zbmath.org/authors/?q=ai:hobanthad.samkhanSummary: Focusing on the ordered semihypergroup, the goal is to find conditions of minimality of left (right) hyperideal, bi-hyperideal and \((0,2)\)-hyperideal in ordered semihypergroups. The study begins by examining basic properties of \((0,2)\)-hyperideal and bi-hyperideal. Using such knowledge demonstrates that if \(A\) is a \(0\)-minimal \((0,2)\)-bi-hyperideal of an ordered semihypergroup \(H\) with zero, then either \((A^2]=\left\{ 0\right\}\) or \(A\) is a left \(0\)-simple.Study on Green's relations in ordered semihypergroupshttps://zbmath.org/1491.060622022-09-13T20:28:31.338867Z"Tang, Jian"https://zbmath.org/authors/?q=ai:tang.jian"Davvaz, Bijan"https://zbmath.org/authors/?q=ai:davvaz.bijanSummary: In this paper, we study the hyper versions of Green's relations in ordered semihypergroups in detail. The Green's relations \(\mathcal{R}\), \(\mathcal{L}\), \(\mathcal{J}\) and \(\mathcal{H}\) in ordered semihypergroups are first introduced, and the relations between them are given. Furthermore, we investigate the properties of Green's relations in ordered semihypergroups. Particularly, we illustrate the Green's relation \(\mathcal{R} \) (resp. \(\mathcal{L}\)) in an ordered semihypergroup \(S\) is not necessarily a left (resp. right) congruence on \(S\) by counterexamples. Meanwhile, we also provide a sufficient condition that makes the above conclusion true. Finally, we introduce the concept of \(a\)-maximal hyperideals of an ordered semihypergroup, and discuss its related properties by terms of the Green's relation \(\mathcal{J}\).Normally conjugative relationshttps://zbmath.org/1491.080012022-09-13T20:28:31.338867Z"Romano, Daniel A."https://zbmath.org/authors/?q=ai:romano.daniel-abrahamInvertible algebras satisfying associative identities with functional variableshttps://zbmath.org/1491.080022022-09-13T20:28:31.338867Z"Movsisyan, Yu."https://zbmath.org/authors/?q=ai:movsisyan.yuri-m"Gevorgyan, A."https://zbmath.org/authors/?q=ai:gevorgyan.a-h|gevorgyan.albert-r|gevorgyan.a-m|gevorgyan.amirjan-lAll maximal idempotent submonoids of generalized cohypersubstitutions of type \(\tau=(2)\)https://zbmath.org/1491.080052022-09-13T20:28:31.338867Z"Chansuriya, Nagornchat"https://zbmath.org/authors/?q=ai:chansuriya.nagornchatSummary: A generalized cohypersubstitution of type \(\tau\) is a mapping \(\sigma\) which maps every \(n_i\)-ary cooperation symbol \(f_i\) to the coterm \(\sigma(f)\) of type \(\tau= (n_i)_{i\in I}\) . Denote by Cohyp\(_G(\tau)\) the set of all generalized cohypersubstitutions of type \(\tau\). Define the binary operation \(o_{CG}\) on Cohyp\(_G(\tau)\) by \(\sigma_1 \circ_{CG}\sigma_2:=\widehat{\sigma}_1\circ\sigma_2\) for all \(\sigma_1,\sigma_2\in\text{Cohyp}_G(\tau)\) and \(\sigma_{id}(f_i):= f_i\) for all \(i\in I\). Then Cohyp\(_G(\tau):= \{\text{Cohyp}_G(\tau)\), \(\circ_{CG},\sigma_{id}\}\) is a monoid. By \textit{N. Saengsura} and \textit{S. Jermjitpornchai} [``Idempotent and regular cohypersubstitutions of type \(\tau = (2)\)'', Int. J. Pure Appl. Math. 86, No. 4 757--766 (2013); \url{doi:10.12732/ijpam.v86i4.13}], the monoid Cohyp\(_G(2)\) was studied. They characterized and presented the idempotent and regular elements of this monoid. In this present paper, we consider the set of all idempotent elements of the monoid Cohyp\(_G(2)\) and determine all maximal idempotent submonoids of this monoid.The Menger algebra of terms induced by order-decreasing transformationshttps://zbmath.org/1491.080102022-09-13T20:28:31.338867Z"Wattanatripop, Khwancheewa"https://zbmath.org/authors/?q=ai:wattanatripop.khwancheewa"Changphas, Thawhat"https://zbmath.org/authors/?q=ai:changphas.thawhatThe authors define the so-called order-decreasing terms of a given similarity type by induction. They prove that the set of these terms is closed under superposition and hence forms a Menger algebra. They continue with hypersubstitutions of order-decreasing terms and related so-called full closed identities and with order-decreasing full closed varieties.
Reviewer: Ivan Chajda (Přerov)Group congruences on trioidshttps://zbmath.org/1491.080112022-09-13T20:28:31.338867Z"Zhuchok, A. V."https://zbmath.org/authors/?q=ai:zhuchok.anatolii-vSummary: We describe all group congruences on an arbitrary trioid and present the least group congruence on a trioid with an inverse semigroup, the least group congruence on a trioid with an orthodox semigroup and the least group congruence on a trioid with a regular semigroup.Structure of relatively free trioidshttps://zbmath.org/1491.080132022-09-13T20:28:31.338867Z"Zhuchok, A. V."https://zbmath.org/authors/?q=ai:zhuchok.anatolii-vSummary: \textit{J.-L. Loday} and \textit{M. Ronco} [Contemp. Math. 346, 369--398 (2004; Zbl 1065.18007)] introduced the notions of a trioid and a trialgebra, and constructed the free trioid of rank \(1\) and the free trialgebra. This paper is a survey of recent developments in the study of free objects in the varieties of trioids and trialgebras. We present the constructions of the free trialgebra and the free trioid, the free commutative trioid, the free \(n\)-nilpotent trioid, the free left (right) \(n\)-trinilpotent trioid, and the free rectangular trioid. Some of these results can be applied to constructing relatively free trialgebras.InfoMod: a visual and computational approach to Gauss' binary quadratic formshttps://zbmath.org/1491.110052022-09-13T20:28:31.338867Z"Zeytin, Ayberk"https://zbmath.org/authors/?q=ai:zeytin.ayberkSummary: InfoMod is a software devoted to the modular group, \(\text{PSL}_2 (\mathbb{Z})\). It consists of algorithms that deal with the classical correspondences among geodesics on the modular surface, elements of the modular group and binary quadratic forms. In addition, the software implements the recently discovered representation of Gauss' indefinite binary quadratic forms and their classes in terms of certain infinite planar graphs (dessins) called çarks. InfoMod illustrates various aspects of these forms, i.e. Gauss' reduction algorithm, the representation problem of forms, ambiguous and reciprocal forms. It can be used for visualization, for high performance computation involving these mathematical structures as well as experimenting.The mathematical artist. A tribute to John Horton Conwayhttps://zbmath.org/1491.110062022-09-13T20:28:31.338867ZPublisher's description: This book brings together the impact of Prof. John Horton Conway, the playful and legendary mathematician's wide range of contributions in science which includes research areas -- Game of Life in cellular automata, theory of finite groups, knot theory, number theory, combinatorial game theory, and coding theory. It contains transcripts where some eminent scientists have shared their first-hand experience of interacting with Conway, as well as some invited research articles from the experts focusing on Game of Life, cellular automata, and the diverse research directions that started with Conway's Game of Life. The book paints a portrait of Conway's research life and philosophical direction in mathematics and is of interest to whoever wants to explore his contribution to the history and philosophy of mathematics and computer science. It is designed as a small tribute to Prof. Conway whom we lost on April 11, 2020.
The articles of this volume will be reviewed individually.Non-commutative methods in additive combinatorics and number theoryhttps://zbmath.org/1491.110172022-09-13T20:28:31.338867Z"Shkredov, Ilya D."https://zbmath.org/authors/?q=ai:shkredov.ilya-dThis work is a survey on the area of arithmetic combinatorics, focussing on non-abelian results. Typically, additive combinatorics have dealt with problems both on the integers on abelian groups, but more recently there have been important breakthroughs on the study of questions in the non-abelian setting. For instance, the growth of sets on non-abelian groups has been a very recent trend of research with several breakthroughs on the last years.
This survey explore a wide variety of results on these area, and relates it with the abelian analogues. These includes problems on arithmetic combinatorics on itself (see for instance Section 4: The structure of sets with small doubling in an arbitrary group), as well as applications on other domains including incidence geometry, group theory and analytic number theory, among other.
Apart from a very detailed survey of the techniques (non-Fourier analysis, Balogh-Szeméredi-Gowers Theorem, etc), the author provides a very rich source of bibliography.
Reviewer: Juanjo Rué Perna (Barcelona)The Eichler integral of \(E_2\) and \(q\)-brackets of \(t\)-hook functionshttps://zbmath.org/1491.110442022-09-13T20:28:31.338867Z"Ono, Ken"https://zbmath.org/authors/?q=ai:ono.kenSummary: For functions \(f: \mathcal{P}\mapsto \mathbb{C}\) on partitions, \textit{S. Bloch} and \textit{A. Okounkov} [Adv. Math. 149, No. 1, 1--60 (2000; Zbl 0978.17016)] defined a power series \(\langle f \rangle_q\) that is the ``weighted average'' of \(f\). As Fourier series in \(q = e^{2 \pi iz}\), such \(q\)-brackets generate the ring of quasimodular forms, and the modular forms that are powers of Dedekind's eta-function. Using work of Berndt and Han, we build modular objects from
\[
f_t(\lambda):= t\sum_{h\in \mathcal{H}_t(\lambda)}\frac{1}{h^2},
\]
weighted sums over partition hook numbers that are multiples of \(t\). We find that \(\langle f_t \rangle_q\) is the Eichler integral of \((1 - E_2(tz))/24\), which we modify to construct a function \(M_t(z)\) that enjoys weight 0 modularity properties. As a consequence, the non-modular Fourier series
\[
H_t^\ast(z):=\sum_{\lambda \in \mathcal{P}} f_t(\lambda)q^{|\lambda |-\frac{1}{24}}
\]
inherits weight \(- 1/2\) modularity properties. These are sufficient to imply a Chowla-Selberg-type result, generalizing the fact that weight \(k\) algebraic modular forms evaluated at discriminant \(D < 0\) points \(\tau\) are algebraic multiples of \(\Omega_D^k\), the \(k\)th power of the canonical period. If we let \(\Psi (\tau):=-\pi i \left (\frac{\tau^2-3\tau +1}{12\tau}\right)-\frac{\log (\tau)}{2}\), then for \(t = 1\) we prove that
\[
\begin{aligned} H_1^\ast(-1/\tau)-\frac{1}{\sqrt{-i\tau}}\cdot H_1^\ast(\tau)\in \overline{\mathbb{Q}}\cdot \frac{\Psi (\tau)}{\sqrt{\Omega_D}}. \end{aligned}
\]
For the entire collection see [Zbl 1479.47003].On length densitieshttps://zbmath.org/1491.130272022-09-13T20:28:31.338867Z"Chapman, Scott T."https://zbmath.org/authors/?q=ai:chapman.scott-thomas"O'Neill, Christopher"https://zbmath.org/authors/?q=ai:oneill.christopher"Ponomarenko, Vadim"https://zbmath.org/authors/?q=ai:ponomarenko.vadimThe set of lengths of an element \(x\) in a commutative monoid \(M\) and the elasticity for elements of monoid \(M\) along with the elasticity of monoid \(M\) itself have been well studied in the literature, particularly for Krull monoid, numerical monoid, Puiseux monoids and arithmetic congruence monoids. This paper discusses various new notions such as a length density \(\operatorname{LD}(x)\), asymptotic length density \(\overline{\operatorname{LD}}(x)\) of an element \(x\) and length density \(\operatorname{LD}(M)\) of a monoid \(M\). Analogous to the acceptance of elasticity of \(M\), the authors define the acceptance of length density. The length density of \(M\) is accepted if there exists \(x\in M\) such that \(\operatorname{LD}(x) = \operatorname{LD}(M)\).
The paper is organised as follows: Introduction includes a brief discussion of all the crucial definitions and notions which are required for the rest of the article. Section 2 discusses the basic properties of length density. In particularly, bounds are calculated for \(\operatorname{LD}(x)\) and \(\operatorname{LD}(M)\)s, and several examples of monoids have been given for which these bounds are met. These results further give examples of monoids for which length density is (is not) accepted. Section 3 answers the problem of the existence of a monoid having any irrational number in \((0, 1)\) as a length density, which is a similar result to the existence of a monoid having irrational elasticity in \((0, 1)\). However, this paper presents a new construction to prove the result for the length density case. Section 3 further computes the length density of block monoids and discusses several other examples. The last section of the article deals with the sufficient condition which guarantees the existence of asymptotic length density, this condition is particularly satisfied by a finitely generated monoid, \(C\)-monoid and Krull monoids with finite divisor class group. Authors also construct an example of a monoid which lacks asymptotic length density.
The article is well readable and has many good results with much information about the literature.
Reviewer: Nitin Bisht (Indore)Application of automorphic forms to lattice problemshttps://zbmath.org/1491.140422022-09-13T20:28:31.338867Z"Düzlü, Samed"https://zbmath.org/authors/?q=ai:duzlu.samed"Krämer, Juliane"https://zbmath.org/authors/?q=ai:kramer.julianeThe paper under review is motivated by the work of de Boer, Ducas, Pellet-Mary, and Wesolowski on self-reducibility of ideal-SVP via Arakelov random walks [\textit{D. Micciancio} (ed.) and \textit{T. Ristenpart} (ed.), Advances in cryptology -- CRYPTO 2020. 40th annual international cryptology conference, CRYPTO 2020. Proceedings. Part II. Cham: Springer. 243--273 (2020; Zbl 07332292)]. More preceisely, the authors show a worst-case to average-case reduction for ideal lattices and explain their approach how the steps are reproduced for module lattices of a fixed rank over some number field. Two major distinctions in their approach are that for higher rank module lattices, the notion of Arakelov divisors is replaced by adèles and Fourier analysis is substituted by the notion of automorphic forms.
Note that subject to the Riemann hypothesis, the worst-case to average-case convergence is analyzed in terms of the Fourier series. Thereafter, the worst-case shortest vector problem is as hard as the averagecase shortest vector problem.
Reviewer: Sami Omar (Sukhair)Symmetric Galois groups under specializationhttps://zbmath.org/1491.140492022-09-13T20:28:31.338867Z"Monderer, Tali"https://zbmath.org/authors/?q=ai:monderer.tali"Neftin, Danny"https://zbmath.org/authors/?q=ai:neftin.dannyConsider a polynomial \(f(t,x) \in \mathbb{Q}(t)[x]\) with coefficients depending on \(t\), and let \(G\) be the Galois group of the polynomial. For all but finitely many specializations \(t \mapsto t_0 \in \mathbb{Q}\), the Galois group \(\text{Gal}(f(t_0,x), \mathbb{Q})\) is a subgroup of \(G\); and for infinitely many \(t_0\), that group is just \(G\). The question in this article is what proper subgroups of \(G\) are \(\text{Gal}(f(t_0,x), \mathbb{Q})\) for infinitely many \(t_0 \in \mathbb{Q}\). The answer to that question is given in the main Theorem 1.1, according to which, if \(G\) is \(A_n\) or \(S_n\) for \(n\) greater than a certain constant \(N_1\), and \(\text{Gal}(f(t_0,x), \mathbb{Q}) \approx H\) for infinitely many \(t_0 \in \mathbb{Q}\), then either i) \(H = A_n\) or \(S_n\), or ii) \(H = A_{n-1}\) or \(S_{n-1}\), or iii) \(A_{n-2} \lneq H \lneq S_{n-2} \times S_2\). The way to prove Theorem 1.1 is to classify low genus covers with monodromy \(A_n\) or \(S_n\). This is made in Theorem 1.2 by an analysis of the transitivity of the action of \(H\) on unordered sets, and Theorem 1.1 follows using Faltings' theorem.
Reviewer: José Javier Etayo (Madrid)Parametrization, structure and Bruhat order of certain spherical quotientshttps://zbmath.org/1491.140752022-09-13T20:28:31.338867Z"Chaput, Pierre-Emmanuel"https://zbmath.org/authors/?q=ai:chaput.pierre-emmanuel"Fresse, Lucas"https://zbmath.org/authors/?q=ai:fresse.lucas"Gobet, Thomas"https://zbmath.org/authors/?q=ai:gobet.thomasLet \(G\) be a connected reductive algebraic group over an algebraically closed field of characteristic zero, let \(B\subset G\) be a Borel subgroup and let \(H=Z_G(e)\) be the isotropy group of a spherical nilpotent orbit \(G.e\subset\mathfrak g\). Recall that a nilpotent orbit \(G.e\) is spherical if and only if \(e\) is a nilpotent element of height less or equal to \(3\). Since the orbit \(G.e\) is here assumed to be spherical, it contains only finitely many \(B\)-orbits as well as the flag variety \(G/B\) has only finitely many \(H\)-orbits.
The authors study the structure of the isotropy group \(H\) and give a parameterization of the \(H\)-orbits in \(G/B\), provided \(e\) is a nilpotent element of height 2, showing that the \(H\)-orbits have the structure of algebraic affine bundles over certain homogeneous \(M\)-spaces, where \(M\) is a Levi subgroup of \(H\). Using their parameterization they introduce a partial order among the \(H\)-orbits and show that in type \(A\) their order coincides with the strong Bruhat order.
Reviewer: Paolo Bravi (Roma)\(N\)-chained semigroups and \(n/2\)-perspective modules and ringshttps://zbmath.org/1491.160072022-09-13T20:28:31.338867Z"Mary, Xavier"https://zbmath.org/authors/?q=ai:mary.xavierLet \(S\) be a semigroup. Two idempotents \(e,f\) are called \textsl{left (resp right) associates} if \(ef=e\) and \(fe=f\) (resp. \(ef=f\), \(fe=e\)), denoted \( e\sim _{l}f\) (resp. \(e\sim _{r}f\)). Two idempotents \(e,f\) are \textsl{ isomorphic}, denoted by \(e\cong f\), if \(e=ab\) and \(f=ba\) for some \(a,b\in S\) . The set of inner (resp. reflexive) inverses of \(a\) will be denoted by \(I(a) \) (resp. \(V(a)\)). Using a notation introduced by Nielsen and independently by Khurana in the context of rings, a sequence of idempotents \( e_{0},\ldots,e_{n}\in S\) such that \(e_{2i}\sim _{r}e_{2i+1}\) and \(e_{2i+1}\sim _{r}e_{2i+2}\), \(0\leq i\leq \lbrack \frac{n-1}{2}]\) is called a \textsl{ right }\(n\)\textsl{-chain}.
Let \(n\in \mathbb{N}\). A semigroup \(S\) is \textsl{left (resp. right) }\(n\) \textsl{-chained} if any two isomorphic idempotents are connected by a left (resp. right) \(n\)-chain. It is \(n\)\textsl{-chained} if its both left and right \(n\)-chained.
In Section 2, the \(n\)-chained semigroups are alternatively be characterized by means of reflexive inverses.
In Section 3, \(n/2\)-perspective modules and rings are defined, studied and related to n-chained semigroups.
For any \(p\in \mathbb{N}\), two direct summands \(A,A^{\prime }\subseteq ^{\oplus }M\) are \(p+1\)\textsl{-perspective} if \(A\sim _{\oplus }^{p}B\sim _{\oplus }A^{\prime }\) for some \(B\subseteq ^{\oplus }M\).
They are \(\pi \)-perspective (power perspective), if they are \(n\)-perspective for some \(n\in \mathbb{N}\).
2. The module \(M\) is \(p\)-perspective, \(p\in \) \(\mathbb{N}\) (resp. \(\pi \) -perspective) if any two isomorphic direct summands are \(p\)-perspective (resp. \(\pi \)-perspective);
3. The module \(M\) is \(p+1/2\)-perspective, \(p\in \mathbb{N}\) if whenever \( M=A\oplus B\) and \(A\cong A^{\prime }\) (\(A,A^{\prime },B\subseteq ^{\oplus }M\) ), then \(M=A^{\prime }\oplus B^{\prime }\) for some \(B^{\prime }\subseteq ^{\oplus }M\) such that \(B\sim _{\oplus }^{p}B^{\prime }\).
We list (just) some of the results.
Theorem 3.5. Let \(M\) be a module, \(R=End(M)\) and \(n\in \mathbb{N}\). Then the following statements are equivalent:
1. \(M\) is \(n/2\)-perspective;
2. the right module \(R_{R}\) (equiv. the left module \(_{R}R\)) is \(n/2\) -perspective;
3. regular endomorphisms of \(M\) are image (equiv. kernel) \(n/2\)-perspective;
4. regular elements of \(\mathcal{M}R\) are \(n+1\)-chained regular (equiv. \(n+1\) -anti-chained regular);
5. The monoid \(\mathcal{M}R\) is right (equiv. left) \(n+1\)-chained. [Here \( \mathcal{M}R\) denotes the multiplicartive monoid of the ring \(R\)].
Section 4 then applies the results to small \(n\in \{0,1,2,3\}\).
Corollary 4.8. Let \(M\) be a module and \(R=End(M)\). The following statements are equivalent:
1. \(M\) is 0-perspective (isomorphic direct summands are equal);
2. \(R\) is abelian;
3. isomorphic idempotents of \(\mathcal{M}R\) are right associates (resp. left associates, resp. equal);
4. regular elements of \(\mathcal{M}R\) are completely regular and right subcommutative (resp. left subcommutative,
resp. subcommutative);
5. direct summands of \(M\) are uniquely complemented;
6. direct summands of \(M\) are fully invariant.
Corollary 4.17. \(R\) is \(1/2\)-perspective if and only if isomorphic idempotents of \(R\) are equal modulo the Jacobson radical.
Proposition 4.20. Let \(M\) be a module, and \(a\in R=End(M)\). Then the following statements are equivalent:
1. \(a\) is image (equiv. kernel) \(3/2\)-perspective;
2. \(a\) is \(4\)-chained regular;
3. \(a\) has a completely regular reflexive inverse (as an element of \( \mathcal{M}R\));
4. \(a\) is special clean (as an element of the ring \(R\)).
Also \(M\) is \(3/2\)-perspective if and only if \(R_{R}\) is \(3/2\)-perspective if and only if \( reg(R)=V(R^{\#})\) if and only if \(\mathcal{M}R\) is \(3/2\)-perspective.
Finally, Section 5 studies \(n/2\)-perspectivity with respect to standard constructions (direct summands and corner rings; lifting idempotents; subrings and factor rings).
Reviewer: Grigore Călugăreanu (Cluj-Napoca)An integral theory of dominant dimension of Noetherian algebrashttps://zbmath.org/1491.160082022-09-13T20:28:31.338867Z"Cruz, Tiago"https://zbmath.org/authors/?q=ai:cruz.tiagoAuthor's abstract: Dominant dimension is introduced into integral representation theory, extending the classical theory of dominant dimension of Artinian algebras to projective Noetherian algebras (that is, algebras which are finitely generated projective as modules over a commutative Noetherian ring). This new homological invariant is based on relative homological algebra introduced by Hochschild in the 1950s. Amongst the properties established here are a relative version of the Morita-Tachikawa correspondence and a relative version of Mueller's characterization of dominant dimension. The behaviour of relative dominant dimension of projective Noetherian algebras under change of ground ring is clarified and we explain how to use this property to determine the relative dominant dimension of projective Noetherian algebras. In particular, we determine the relative dominant dimension of Schur algebras and quantized Schur algebras.
Reviewer: Udhayakumar Ramalingam (Vellore)Locally solvable subnormal and quasinormal subgroups in division ringshttps://zbmath.org/1491.160202022-09-13T20:28:31.338867Z"Le Qui Danh"https://zbmath.org/authors/?q=ai:le-qui-danh."Huynh Viet Khanh"https://zbmath.org/authors/?q=ai:huynh-viet-khanh.Summary: In this paper, we show that every locally solvable subnormal subgroup or locally solvable quasinormal subgroup of the multiplicative group of a division ring is central.On the ideals of the Radford Hopf algebrashttps://zbmath.org/1491.160342022-09-13T20:28:31.338867Z"Wang, Yu"https://zbmath.org/authors/?q=ai:wang.yu.1|wang.yu.9|wang.yu.3|wang.yu.4|wang.yu.8|wang.yu.2|wang.yu|wang.yu.5"Zheng, Ying"https://zbmath.org/authors/?q=ai:zheng.ying"Li, Libin"https://zbmath.org/authors/?q=ai:li.libinThis paper is devoted to the study of Radford Hopf algebra, which is introduced by Radford in order to give an example of Hopf algebra whose Jacobson radical is not a Hopf ideal. In the introduction part, the authors give a brief survey on the historic study of rings and enveloping algebras which have all ideals principal. The main result of this paper is the following: the Radford Hopf algebra \(H_{m, n}\) is a principal ideal ring. The authors also give explicitly the generators of all \(28\) ideals of \(9\)-dimensional Radford Hopf algebra \(H_{1, 3}.\)
Reviewer: Tongsuo Wu (Shanghai)ChevLie: constructing Lie algebras and Chevalley groupshttps://zbmath.org/1491.170012022-09-13T20:28:31.338867Z"Geck, Meinolf"https://zbmath.org/authors/?q=ai:geck.meinolfSummary: We present \(\mathsf{ChevLie-1.1}\), a module for \(\mathsf{Julia}\) and, ultimately, the emerging \(\mathsf{OSCAR}\) system. It provides functions for constructing simple Lie algebras and the corresponding Chevalley groups (of adjoint or other types), using a recently established approach via Lusztig's ``canonical bases''. These programs, combined with the \(\mathsf{Julia}\) interface to \(\mathsf {SINGULAR}\), supply an efficient, user-friendly way to establish a key part of a new characterisation of Lusztig's ``special'' nilpotent orbits in simple Lie algebras.On a conjecture about an analogue of Tokuyama's theorem for \(G_2\)https://zbmath.org/1491.170042022-09-13T20:28:31.338867Z"DeFranco, Mario"https://zbmath.org/authors/?q=ai:defranco.mario\textit{T. Tokuyama} [J. Math. Soc. Japan 40, No. 4, 671--685 (1988; Zbl 0639.20022)] expressed the product of the deformed Weyl denominator and a Schur polynomial as a sum over strict Gelfand-Tsetlin patterns. This may be recast using crystal graphs as follows (see [\textit{B. Brubaker} et al., Weyl group multiple Dirichlet series. Type A combinatorial theory. Princeton, NJ: Princeton University Press (2011; Zbl 1288.11052)], Chapter 5). Let \(\lambda\in \mathfrak{sl}_{r+1}(\mathbb{C})\) be a dominant weight, \(\chi_\lambda\) the character of the associated representation of highest weight \(\lambda\), and \(\rho\) be the Weyl vector. Then Tokuyama's formula is an expression for the product of \(\chi_\lambda\) with the deformed Weyl denominator as a sum over \(\mathcal{B}_{\lambda+\rho}\), the Kashiwara crystal with highest weight \(\lambda+\rho\). There is a substantial history of this formula for other Cartan types, explained in the paper under review. In particular, \textit{H. Friedlander} et al. [J. Algebr. Comb. 41, No. 4, 1089--1102 (2015; Zbl 1316.05128)] conjectured such a formula for the exceptional Lie algebra of type \(G_2\). The paper under review establishes this conjecture using combinatorial methods. The author expresses both sides of the conjectured identity as polynomials in four variables whose coefficients are rational functions, and then shows that these coefficients are equal.
A complement to this work is the paper of \textit{S. Leslie} [Sel. Math., New Ser. 25, No. 3, Paper No. 41, 50 p. (2019; Zbl 1491.17005)], which gives a different expression in type \(G_2\) for the character times the deformed Weyl denominator, namely a sum over \(\mathcal{B}_{\lambda+\rho}\) plus a geometric error term coming from finitely many crystals of resonant Mirković-Vilonen polytopes.
Reviewer: Solomon Friedberg (Chestnut Hill)Resonant Mirković-Vilonen polytopes and formulas for highest-weight charactershttps://zbmath.org/1491.170052022-09-13T20:28:31.338867Z"Leslie, Spencer"https://zbmath.org/authors/?q=ai:leslie.spencerSummary: Formulas for the product of an irreducible character \(\chi _\lambda \) of a complex Lie group and a deformation of the Weyl denominator as a sum over the crystal \({\mathcal {B}}(\lambda +\rho )\) go back to Tokuyama. We study the geometry underlying such formulas using the expansion of spherical Whittaker functions of \(p\)-adic groups as a sum over the canonical basis \({\mathcal {B}}(-\infty )\), which we show may be understood as arising from tropicalization of certain toric charts that appear in the theory of total positivity and cluster algebras. We use this to express the terms of the expansion in terms of the corresponding Mirković-Vilonen polytope. In this non-archimedean setting, we identify resonance as the appropriate analogue of total positivity, and introduce \textit{resonant Mirković-Vilonen polytopes} as the corresponding geometric context. Focusing on the exceptional group \(G_2\), we show that these polytopes carry new crystal graph structures which we use to compute a new Tokuyama-type formula as a sum over \({\mathcal {B}}(\lambda +\rho )\) plus a geometric error term coming from finitely many crystals of resonant polytopes.Quasi-split symmetric pairs of \(U(\mathfrak{gl}_N)\) and their Schur algebrashttps://zbmath.org/1491.170072022-09-13T20:28:31.338867Z"Li, Yiqiang"https://zbmath.org/authors/?q=ai:li.yiqiang|li.yiqiang.1"Zhu, Jieru"https://zbmath.org/authors/?q=ai:zhu.jieruA classical result in representation theory states that the action of the symmetric group \(S_d\) fully centralizes the natural action of the complex general linear algebra \(\mathfrak{gl}_N\) on the tensor space \((\mathbb C^N)^{\otimes d}\). This results in representations for \(\mathfrak{gl}_N\), which are summands of \((\mathbb C^N)^{\otimes d}\), being in bijection with representations for the symmetric group \(S_d\). The Schur algebra of type A is the centralizer algebra of \(S_d\) on \((\mathbb{C}^N)^{\otimes d}\).
In type B, the orthogonal group does not fully centralize the action of the type B Weyl group on the tensor space. The orthogonal group is known to centralize the action of the Brauer algebra but the description of the centralizer of the type B Weyl group action, and its quantization, is nontrivial. In one approach, the centralizer is given by a subgroup of \(GL_N\), its Lie algebra being the fixed-point subalgebra \(\mathfrak{gl}_N^{\theta}\) of \(\mathfrak{gl}_N\) under a certain involution \(\theta\). In another approach, the centralizer is given as a homomorphic image of a two-block subalgebra \(U\) of \(U(\mathfrak{gl}_N)\). The pairs \((U(\mathfrak{gl}_N), U)\) and \((U(\mathfrak{gl}_N), U(\mathfrak{gl}_N^{\theta}))\) are infinitesimal quasi-split symmetric pairs of type A.
Y. Li and J. Zhu establish explicit isomorphisms between \(U\) and \(U(\mathfrak{gl}_N^{\theta})\) (Theorem 2.4.2, page 14), and consequently on their respective Schur algebras. The authors also provide a presentation of the geometric counterpart of the above Schur algebras specialized at \(q=1\).
Reviewer: Mee Seong Im (Annapolis)A class of non-weight modules of \(U_p(\mathfrak{s} \mathfrak{l}_2)\) and Clebsch-Gordan type formulashttps://zbmath.org/1491.170082022-09-13T20:28:31.338867Z"Cai, Yan-an"https://zbmath.org/authors/?q=ai:cai.yanan"Chen, Hongjia"https://zbmath.org/authors/?q=ai:chen.hongjia"Guo, Xiangqian"https://zbmath.org/authors/?q=ai:guo.xiangqian"Ma, Yao"https://zbmath.org/authors/?q=ai:ma.yao"Zhu, Mianmian"https://zbmath.org/authors/?q=ai:zhu.mianmianQuantum groups were initially constructed by Drinfeld, and independently by Jimbo, to study (quantum) Yang-Baxter equations. Given a finite-dimensional semisimple Lie algebra \(\mathfrak{g}\), the quantum group \(U_q(\mathfrak{g})\) is a Hopf algebra that depends on a parameter \(q\in \mathbb{C}\setminus \{0\}\), which is a deformation of the universal enveloping algebra \(U(\mathfrak{g})\).
The present authors construct a class of modules for the quantum group \(U_q(\mathfrak{sl}_2)\), which is free of rank \(1\) when restricted to \(\mathbb{C}[K^{\pm 1}]\); it corresponds to the universal enveloping algebra of the Cartan subalgebra of \(\mathfrak{sl}_2\). They prove that any \(\mathbb{C}[K^{\pm 1}]\)-free \(U_q(\mathfrak{sl}_2)\)-module of rank \(1\) is isomorphic to one of the modules they constructed (Theorem 3.2, page 747), and their isomorphism classes are obtained (Corollary 3.3, page 748). They also investigate the tensor products of the \(\mathbb{C}[K^{\pm 1}]\)-free modules with finite-dimensional simple modules over \(U_q(\mathfrak{sl}_2)\). For the generic cases, they obtain direct sum decomposition formulas for them, which are similar to the well-known Clebsch-Gordan formula for tensor products between finite-dimensional weight modules over \(U_q(\mathfrak{sl}_2)\) (Theorem 4.2, page 750).
Reviewer: Mee Seong Im (Annapolis)Classification on irreducible Whittaker modules over quantum group \(U_q(\mathfrak{sl}_3,\Lambda)\)https://zbmath.org/1491.170122022-09-13T20:28:31.338867Z"Xia, Limeng"https://zbmath.org/authors/?q=ai:xia.limeng"Guo, Xiangqian"https://zbmath.org/authors/?q=ai:guo.xiangqian"Zhang, Jiao"https://zbmath.org/authors/?q=ai:zhang.jiaoWhittaker modules for a simple finite-dimensional complex Lie algebra \(\mathfrak{g}\) have close connections with the center \(Z\) of \(U(\mathfrak{g})\). That is, there is a bijection between the set of all equivalence classes of Whittaker modules and the set of all ideals in \(Z\), and irreducible Whittaker modules correspond to maximal ideals of \(Z\) under this bijection.
Whittaker modules for \(U_q(\mathfrak{sl}_2)\) have been extensively studied, but for \(U_q(\mathfrak{g})\), where \(\mathfrak{g}\not= \mathfrak{sl}_2\), there is an obstruction to construct Whittaker modules. The subalgebras \(U_q(\mathfrak{n}_+)\subset U_q(\mathfrak{g})\), for \(\mathfrak{g}\not= \mathfrak{sl}_2\), generated by positive root vectors and subject to the quantum Serre relations do not have nonsingular characters. Savostyanov overcame this obstruction by considering the topological Hopf algebra \(U_h(\mathfrak{g})\) over \(\mathbb{C}[[h]]\). He used the Coxeter realization \(U_h^{s_{\pi}}(\mathfrak{g})\) of the quantum group \(U_h(\mathfrak{g})\) corresponding to the Coxeter element \(s_{\pi}\) and showed that the subalgebras \(U_h^{s_{\pi}}(\mathfrak{n}_+)\subset U_h^{s_{\pi}}(\mathfrak{g})\) generated by positive root vectors do indeed possess nonsingular characters.
The present authors define the Whittaker modules over the simply-connected quantum group \(U_q(\mathfrak{sl}_3,\Lambda)\), where \(\Lambda\) is the weight lattice of the Lie algebra \(\mathfrak{sl}_3\), and then they completely classify all those simple ones. That is, a simple Whittaker module over \(U_q(\mathfrak{sl}_3,\Lambda)\) is either a highest weight module, or determined by two parameters \(z\in \mathbb{C}\) and \(\gamma\in \mathbb{C}^*\), up to a Hopf automorphism (see Theorem 1, page 1091, and Theorem 2, page 1092).
Reviewer: Mee Seong Im (Annapolis)A survey of selected categorical properties of algebras with multivalued additionhttps://zbmath.org/1491.180042022-09-13T20:28:31.338867Z"Borczyk, Wojciech"https://zbmath.org/authors/?q=ai:borczyk.wojciech"Gładki, Paweł"https://zbmath.org/authors/?q=ai:gladki.pawel"Worytkiewicz, Krzysztof"https://zbmath.org/authors/?q=ai:worytkiewicz.krzysztofThis survey paper reviews quite-known results on selected categorical properties of module-like objects with multivalued addition. A multivalued operation on a set \(A\) is a function from \(A\times A\) into \(P^*(A)\), the set of non-empty subsets of \(A\).
First, the authors begin with listing all familiar definitions of algebras endowed with hyperoperations, e.g., \(H_v\)-semigroup, hypergroup, polygroup, canonical hypergroup, \(H_v\)-ring, hyperring, Krasner hyperring, \(H_v\)-module, \(H\)-hypermodule over a fixed Krasner hyperring \(H\), and then reduce their considerations to a few categories of objects which are similar to modules with a few different notions of morphisms. They then discuss various characterizations of monics and epics in these categories. In the next sections, they focus on products, coproducts, free objects, subobjects and quotient objects. Finally, they discuss equalizers and kernels, as well as coequalizers and cokernels, and conclude this survey with a result stating that the category of hypermodules with strong homomorphisms (possibly multivalued--see the paper for precise definitions) is Puppe exact, that is, it has zero objects, kernels and cokernels, is normal and conormal, and such that every morphism admits an epi-mono factorization.
Reviewer's remark: In seems that the Proposition 6.2 is not valid; equalizers do not exist in the category left hypermodules over a Krasner hyperring \(H\) with strong homomorphisms. For a related discussion on hypergroups see [\textit{Z. Kanjanzadeh} and \textit{A. Madanshekaf}, Turk. J. Math. 43, No. 5, 2578--2591 (2019; Zbl 1429.20051)].
For the entire collection see [Zbl 1460.13001].
Reviewer: Ali Madanshekaf (Semnan)Third homology of perfect central extensionshttps://zbmath.org/1491.190032022-09-13T20:28:31.338867Z"Mirzaii, B."https://zbmath.org/authors/?q=ai:mirzaii.behrooz"Mokari, F. Y."https://zbmath.org/authors/?q=ai:mokari.fatemeh-y"Ordinola, D. C."https://zbmath.org/authors/?q=ai:ordinola.d-cGiven a central extension \(1\to A\to G\to Q\to1\), the authors prove results about image of the map \(H_3(A,\mathbb{Z})\to H_3(G,\mathbb{Z})\) and the kernel and cokernel of \(H_3(G,\mathbb{Z})\to H_3(Q,\mathbb{Z})\) under certain additional hypotheses. They study the cohomology map \(H^3(Q,\mathbb{Z})\to H^3(G,\mathbb{Z})\) under certain hypotheses. They also provide an application to the third Hermitian \(K\)-theory of rings with involution.
Reviewer: Kevin Hutchinson (Dublin)An introduction to abstract algebra. Sets, groups, rings, and fieldshttps://zbmath.org/1491.200012022-09-13T20:28:31.338867Z"Weintraub, Steven H."https://zbmath.org/authors/?q=ai:weintraub.steven-hPublisher's description: This book is a textbook for a semester-long or year-long introductory course in abstract algebra at the upper undergraduate or beginning graduate level.
It treats set theory, group theory, ring and ideal theory, and field theory (including Galois theory), and culminates with a treatment of Dedekind rings, including rings of algebraic integers.
In addition to treating standard topics, it contains material not often dealt with in books at this level. It provides a fresh perspective on the subjects it covers, with, in particular, distinctive treatments of factorization theory in integral domains and of Galois theory.
As an introduction, it presupposes no prior knowledge of abstract algebra, but provides a well-motivated, clear, and rigorous treatment of the subject, illustrated by many examples. Written with an eye toward number theory, it contains numerous applications to number theory (including proofs of Fermat's theorem on sums of two squares and of the Law of Quadratic Reciprocity) and serves as an excellent basis for further study in algebra in general and number theory in particular.
Each of its chapters concludes with a variety of exercises ranging from the straightforward to the challenging in order to reinforce students' knowledge of the subject. Some of these are particular examples that illustrate the theory while others are general results that develop the theory further.On an analogue of \(L^2\)-Betti numbers for finite field coefficients and a question of Atiyahhttps://zbmath.org/1491.200022022-09-13T20:28:31.338867Z"Neumann, Johannes"https://zbmath.org/authors/?q=ai:neumann.johannesSummary: We construct a dimension function for modules over the group ring of an amenable group. This may replace the von Neumann dimension in the definition of \(L^2\)-Betti numbers and thus allows an analogous definition for finite field coefficients. Furthermore we construct examples for characteristic 2 in answer to Atiyah question of irrational \(L^2\)-Betti numbers.Elements of quasigroup theory and applicationshttps://zbmath.org/1491.200032022-09-13T20:28:31.338867Z"Shcherbacov, Victor A."https://zbmath.org/authors/?q=ai:shcherbacov.victor-aPublisher's description: This book provides an introduction to quasigroup theory along with new structural results on some of the quasigroup classes. Many results are presented with some of them from mathematicians of the former USSR. These included results have not been published before in the western mathematical literature. In addition, many of the achievements obtained with regard to applications of quasigroups in coding theory and cryptology are described.The Sylow theoremshttps://zbmath.org/1491.200042022-09-13T20:28:31.338867Z"Hauchecorne, Bertrand"https://zbmath.org/authors/?q=ai:hauchecorne.bertrandSummary: La genèse de la théorie des groupes est en général associée aux écrits de Galois, en 1832, juste avant son décès dans un tragique duel. On peut s'étonner que l'un des théorèmes les plus simples de cette théorie porte le nom de Lagrange, savant disparu près de vingt ans plus tôt. En fait, l'histoire des groupes de substitutions, on préfère dire aujourd'hui permutations, a commencé bien avant Galois.Nearly linear time isomorphism algorithms for some nonabelian group classeshttps://zbmath.org/1491.200052022-09-13T20:28:31.338867Z"Das, Bireswar"https://zbmath.org/authors/?q=ai:das.bireswar"Sharma, Shivdutt"https://zbmath.org/authors/?q=ai:sharma.shivduttSummary: The Isomorphism problem for groups, when the groups are given by their Cayley tables is a well-studied problem. This problem has been studied for various restricted classes of groups. \textit{T. Kavitha} gave a linear time isomorphism algorithm for abelian groups [J. Comput. Syst. Sci. 73, No. 6, 986--996 (2007; Zbl 1165.68036)]. Although there are isomorphism algorithms for certain nonabelian group classes, the complexities of those algorithms are usually super-linear. In this paper, we design linear and nearly linear time isomorphism algorithms for some nonabelian groups. More precisely,
\begin{itemize}
\item We design a linear time algorithm to factor Hamiltonian groups. This allows us to obtain an \(\mathcal{O}(n)\) algorithm for the Isomorphism problem of Hamiltonian groups, where \(n\) is the order of the groups.
\item We design a nearly linear time algorithm to find a maximal abelian factor of an input group. As a byproduct we obtain an \(\tilde{\mathcal{O}}(n)\) isomorphism for groups that can be decomposed as a direct product of a nonabelian group of bounded order and an abelian group, where \(n\) is the order of the groups.
\end{itemize}
For the entire collection see [Zbl 1416.68013].Finding intermediate subgroupshttps://zbmath.org/1491.200062022-09-13T20:28:31.338867Z"Hulpke, Alexander"https://zbmath.org/authors/?q=ai:hulpke.alexanderSummary: This article describes a practical approach for determining the lattice of subgroups \(U<V<G\) between given subgroups \(U\) and \(G\), provided the total number of such subgroups is not too large. It builds on existing functionality for element conjugacy, double cosets and maximal subgroups.Heptavalent arc-transitive Cayley graphs of Frobenius groups with soluble vertex stabilizerhttps://zbmath.org/1491.200072022-09-13T20:28:31.338867Z"Liu, Harlin"https://zbmath.org/authors/?q=ai:liu.harlinSummary: A Cayley graph \(\Gamma\) is said to be arc-transitive if its full automorphism group \(\Aut\Gamma\) is transitive on the arc set of \(\Gamma\). In this paper, we give a characterization of heptavalent arc-transitive Cayiey graphs on a class of Frobenius groups with soluble vertex stabilizer.Random generation of permutations of the symmetric group or the alternating grouphttps://zbmath.org/1491.200082022-09-13T20:28:31.338867Z"Freyre, P."https://zbmath.org/authors/?q=ai:freyre.p"Díaz, N."https://zbmath.org/authors/?q=ai:diaz.nicole|diaz.nahimSummary: In this paper three new algorithms for the random generation of permutations of degree \(n\) are shown, two of them for the Symmetric group \(S_n\) and one for the Alternating group \(A_n\), in such a way, that they are mainly carried out by means of the operation of additions \(\mod\) \(n\) \(o\) \(\mod\) \(i\), \(i = n..2\).Maximal abelian subgroups of the finite symmetric grouphttps://zbmath.org/1491.200092022-09-13T20:28:31.338867Z"Konieczny, Janusz"https://zbmath.org/authors/?q=ai:konieczny.januszSummary: Let \(G\) be a group. For an element \(a\in G\), denote by \(C^2(a)\) the second centralizer of \(a\) in \(G\), which is the set of all elements \(b\in G\) such that \(bx=xb\) for every \(x\in G\) that commutes with \(a\). Let \(M\) be any maximal abelian subgroup of \(G\). Then \(C^2(a)\subseteq M\) for every \(a\in M\). The \textit{abelian rank} (\textit{\(a\)-rank}) of \(M\) is the minimum cardinality of a set \(A\subseteq M\) such that \(\bigcup_{a\in A} C^2(a)\) generates \(M\). Denote by \(S_n\) the symmetric group of permutations on the set \(X=\{1,\dots,n\}\). The aim of this paper is to determine the maximal abelian subgroups of \(S_n\) of \(a\)-rank 1 describe a class of maximal abelian subgroups of \(S_n\) of \(a\)-rank at most 2.Computing the dimension of ideals in group algebras, with an application to coding theoryhttps://zbmath.org/1491.200102022-09-13T20:28:31.338867Z"Elia, Michele"https://zbmath.org/authors/?q=ai:elia.michele"Gorla, Elisa"https://zbmath.org/authors/?q=ai:gorla.elisa(no abstract)On the primitive irreducible representations of finitely generated linear groups of finite rankhttps://zbmath.org/1491.200112022-09-13T20:28:31.338867Z"Tushev, A. V."https://zbmath.org/authors/?q=ai:tushev.anatolii-vOn the semisimplicity of the cyclotomic quiver Hecke algebra of type \(C\)https://zbmath.org/1491.200122022-09-13T20:28:31.338867Z"Speyer, Liron"https://zbmath.org/authors/?q=ai:speyer.lironIn this paper, the author provides a criteria for the cyclotomic quiver Hecke algebras of type \(C\) to be semisimple. In the semisimple case, the author constructs the irreducible modules.
Reviewer: Hu Jun (Beijing)Recognition of Janko groups and some simple \(K_4\)-groups by the order and one irreducible character degree or character degree graphhttps://zbmath.org/1491.200132022-09-13T20:28:31.338867Z"Behravesh, Hoshang"https://zbmath.org/authors/?q=ai:behravesh.hoshang"Ghaffarzadeh, Mehdi"https://zbmath.org/authors/?q=ai:ghaffarzadeh.mehdi"Ghasemi, Mohsen"https://zbmath.org/authors/?q=ai:ghasemi.mohsen"Hekmatara, Somayeh"https://zbmath.org/authors/?q=ai:hekmatara.somayehSummary: In this paper we prove that some Janko groups are uniquely determined by their orders and one irreducible character degree. Also we prove that some finite simple \(K_4\)-groups are uniquely determined by their character degree graphs and their orders.Representability of permutation representations on coalgebras and the isomorphism problemhttps://zbmath.org/1491.200142022-09-13T20:28:31.338867Z"Costoya, Cristina"https://zbmath.org/authors/?q=ai:costoya.cristina"Méndez, David"https://zbmath.org/authors/?q=ai:mendez.david"Viruel, Antonio"https://zbmath.org/authors/?q=ai:viruel.antonioSummary: Let \(H\) be an arbitrary group and let \(\rho :H\rightarrow \mathrm{Sym}(V)\) be any permutation representation of \(H\) on a set \(V\). We prove that there is a faithful \(H\)-coalgebra \(C\) such that \(H\) arises as the image of the restriction of \(\mathrm{Aut} (C)\) to \(G(C)\), the set of grouplike elements of \(C\). Furthermore, we show that \(V\) can be regarded as a subset of \(G(C)\) invariant under the \(H\)-action and that the composition of the inclusion \(H\hookrightarrow \mathrm{Aut}(C)\) with the restriction \(\mathrm{Aut} (C)\rightarrow \mathrm{Sym}(V)\) is precisely \(\rho\). We use these results to prove that isomorphism classes of certain families of groups can be distinguished through the coalgebras on which they act faithfully.The character table of a sharply 5-transitive subgroup of the alternating group of degree 12https://zbmath.org/1491.200152022-09-13T20:28:31.338867Z"Gill, Nick"https://zbmath.org/authors/?q=ai:gill.nick"Hughes, Sam"https://zbmath.org/authors/?q=ai:hughes.samSummary: We calculate the character table of a sharply \(5\)-transitive subgroup of \(\mathrm{Alt}(12)\), and of a sharply \(4\)-transitive subgroup of \(\mathrm{Alt}(11)\). Our presentation of these calculations is new because we make no reference to the sporadic simple Mathieu groups, and instead deduce the desired character tables using only the existence of the stated multiply transitive permutation representations.Subgroups of arbitrary even ordinary depthhttps://zbmath.org/1491.200162022-09-13T20:28:31.338867Z"Janabi, Hayder"https://zbmath.org/authors/?q=ai:janabi.hayder-abbas"Breuer, Thomas"https://zbmath.org/authors/?q=ai:breuer.thomas"Horváth, Erzsébet"https://zbmath.org/authors/?q=ai:horvath.erzsebetSummary: We show that for each positive integer \(n\), there exist a group \(G\) and a subgroup \(H\) such that the ordinary depth \(d(H,G)\) is \(2n\). This solves the open problem posed by Lars Kadison whether even ordinary depth larger than \(6\) can occur.Kernels of \(p'\)-degree irreducible charactershttps://zbmath.org/1491.200172022-09-13T20:28:31.338867Z"Moretó, Alexander"https://zbmath.org/authors/?q=ai:moreto.alexander"Rizo, Noelia"https://zbmath.org/authors/?q=ai:rizo.noeliaSummary: Let \(G\) be a finite group and let \(p\) be a prime number. We prove that if \(\chi\in\operatorname{Irr}_{p'}(G)\) and \(\operatorname{Ker}\chi\) does not have a solvable normal \(p\)-complement then there exists \(\psi\in\operatorname{Irr}_{p'}(G)\) such that \(\psi(1) > \chi(1)\) and \(\operatorname{Ker}\psi < \operatorname{Ker}\chi\). This is a \(p'\)-version of a classical theorem of Broline and Garrison. As a consequence, we obtain results on \(p\)-parts of character codegrees.On a two-fold cover \(2.(2^6 \cdot G_2 (2))\) of a maximal subgroup of Rudvalis group \(Ru\)https://zbmath.org/1491.200182022-09-13T20:28:31.338867Z"Prins, Abraham Love"https://zbmath.org/authors/?q=ai:prins.abraham-loveSummary: The Schur multiplier \(M(\overline{G_1}) \cong 4\) of the maximal subgroup \(\overline{G_1} = 2^6 \cdot G_2(2)\) of the Rudvalis sporadic simple group Ruis a cyclic group of order 4. Hence a full representative group \(R\) of the type \(R = 4.(2^6 \cdot G_2(2))\) exists for \(\overline{G_1}\). Furthermore, \(\overline{G_1}\) will have four sets \(\operatorname{IrrProj}(\overline{G_1}, \alpha_i)\) of irreducible projective characters, where the associated factor sets \(\alpha_1, \alpha_2, \alpha_3\) and \(\alpha_4\), have orders of 1, 2, 4 and 4, respectively. In this paper, we will deal with a 2-fold cover \(2. \overline{G_1}\) of \(\overline{G_1}\) which can be treated as a non-split extension of the form \(\overline{G_1}= 2^7 \cdot G_2(2)\). The ordinary character table of Ḡwill be computed using the technique of the so-called Fischer matrices. Routines written in the computer algebra system GAP will be presented to compute the conjugacy classes and Fischer matrices of \(\overline{G_1}\) and as well as the sizes of the sets \(|\operatorname{IrrProj}(H_i, \alpha_i)|\)associated with each inertia factor Hi. From the ordinary irreducible characters \(\operatorname{Irr}(\overline{G_1})\) of \(\overline{G_1}\), the set \(\operatorname{IrrProj}(\overline{G_1}, \alpha_2)\) of irreducible projective characters of \(\overline{G_1}\) with factor set \(\alpha_2\) such that \(\alpha^2_2 = 1\), can be obtained.The projective character tables of the maximal subgroups of the Mathieu groups \(M_{23}\) and \(M_{24}\)https://zbmath.org/1491.200192022-09-13T20:28:31.338867Z"Prins, Abraham Love"https://zbmath.org/authors/?q=ai:prins.abraham-loveSummary: It is well known that all the irreducible projective characters of a finite group \(G\) can be obtained from the ordinary irreducible characters of a so-called representation group \(R=M(G)\).\(G\) of \(G\), where \(M(G)\) denotes the Schur multiplier of \(G\). Using this theory, a routine written in the computational algebra system GAP is presented to compute the irreducible projective characters IrrProj \((G, \alpha_i)\) with associated factor sets \(\alpha_i\) for all of the maximal subgroups of the sporadic simple Mathieu groups \(M_{23}\) and \(M_{24}\). In fact, this routine can be applied to any finite group \(G\) provided the ordinary irreducible characters of a representation group \(R\) of \(G\) can be found.\(p\)-blocks relative to a character of a normal subgroup. IIhttps://zbmath.org/1491.200202022-09-13T20:28:31.338867Z"Rizo, Noelia"https://zbmath.org/authors/?q=ai:rizo.noeliaSummary: Let \(p\) be a prime number, let \(G\) be a finite group, let \(N\) be a normal subgroup of \(G\), and let \(\theta\) be a \(G\)-invariant irreducible character of \(N\). In [the author, J. Algebra 514, 254--272 (2018; Zbl 1446.20019)], we introduced a canonical partition of the set \(\mathrm{Irr}(G|\theta)\) of irreducible constituents of the induced character \(\theta^G\), relative to the prime \(p\). We call the elements of this partition the \(\theta\)-blocks. In this paper, we construct a canonical basis of the complex space of class functions defined on \(\{ x \in G \mid x_p \in N\}\), which supersedes previous non-canonical constructions. This allows us to define \(\theta \)-decomposition numbers in a natural way. We also prove that the elements of the partition of \({\mathrm{Irr}}(G|\theta)\) established by these \(\theta \)-decomposition numbers are the \(\theta\)-blocks.Linear \(p\)-Brauer characters and \(p\)-blockshttps://zbmath.org/1491.200212022-09-13T20:28:31.338867Z"Chen, Xiaoyou"https://zbmath.org/authors/?q=ai:chen.xiaoyou"Zeng, Jiwen"https://zbmath.org/authors/?q=ai:zeng.jiwen(no abstract)On Ext-quivers of blocks of weight two for symmetric groupshttps://zbmath.org/1491.200222022-09-13T20:28:31.338867Z"Danz, Susanne"https://zbmath.org/authors/?q=ai:danz.susanne"Erdmann, Karin"https://zbmath.org/authors/?q=ai:erdmann.karinIn representation theory of finite groups, one of the most interesting and important problems should be describing \(p\)-blocks (block algebras) \(B\) of finite groups \(G\) up to Morita equivalence, here \(p\) is a prime that is the characteristic of the ground field \(F\). Surely, Morita equivalence is completely determined by the Ext-quiver and its relations due to P.~Gabriel. This closely related to (so-called) Donovan's conjecture presented as Conjecture M [\textit{J. L. Alperin}, Proc. Symp. Pure Math. 37, 369--375 (1980; Zbl 0449.20019)]. In the article under review, the authors treat with the problem determining algebra structure of the \(p\)-block \(B\) in the case where the finite group \(G\) is the symmetric group \({\mathfrak S}_n\) and the ground field \(F\) has characteristic \(p\geq 3\), and most importantly further in the case that the block \(B\) is of (\(p\)-)weight \(w=2\). It turns out that a defect group \(D\) of \(B\) is actually elementary abelian of order \(p^2\), that is the smallest case in wild-representation type. One of the breakthrough on modular representation theory of symmetric groups goes back to the celebrated paper of \textit{J. Scopes} [J. Algebra 142, No. 2, 441--455 (1991; Zbl 0736.20008)]. Finally, their main theorems say that for given two \(p\)-blocks \(B_1\) and \(B_2\) of symmetric groups of weight \(2\) such that their \(p\)-cores are hook partitions, \(B_1\) and \(B_2\) are Morita equivalent if and only if \(B_2\) or its conjugate block is in the same Scopes class as \(B_1\).
Reviewer: Shigeo Koshitani (Chiba)On the moduli spaces of commuting elements in the projective unitary groupshttps://zbmath.org/1491.200232022-09-13T20:28:31.338867Z"Adem, Alejandro"https://zbmath.org/authors/?q=ai:adem.alejandro"Cheng, Man Chuen"https://zbmath.org/authors/?q=ai:cheng.man-chuenSummary: We provide descriptions for the moduli spaces $\mathrm{Rep}(\Gamma, PU(m))$, where $\Gamma$ is any finitely generated abelian group and $PU(m)$ is the group of \(m \times m\) projective unitary matrices. As an application, we show that for any connected CW-complex \(X\) with $\pi_1(X) \cong \mathbb{Z}^n$, the natural map $\pi_{0}(\mathrm{Rep}(\pi_{1}(X), PU(m))) \rightarrow [X, BPU(m)]$ is injective, hence providing a complete enumeration of the isomorphism classes of flat principal $PU(m)$-bundles over \(X\).
{\par\copyright 2019 American Institute of Physics}A survey on projectively equivalent representations of finite groupshttps://zbmath.org/1491.200242022-09-13T20:28:31.338867Z"Costache, Tania Luminaţa"https://zbmath.org/authors/?q=ai:costache.tania-luminataSummary: The paper is a survey type article in which we present some results on projectively equivalent representations of finite groups.Induced operators on the generalized symmetry classes of tensorshttps://zbmath.org/1491.200252022-09-13T20:28:31.338867Z"Rafatneshan, Gholamreza"https://zbmath.org/authors/?q=ai:rafatneshan.gholamreza"Zamani, Yousef"https://zbmath.org/authors/?q=ai:zamani.yousefSummary: Let \(V\) be a unitary space. Suppose \(G\) is a subgroup of the symmetric group of degree \(m\) and \(\Lambda\) is an irreducible unitary representation of \(G\) over a vector space \(U\). Consider the generalized symmetrizer on the tensor space \(U\otimes V^{\otimes m}\),
\[
S_\Lambda (u\otimes v^\otimes)=\dfrac{1}{|G|} \sum_{\sigma\in G} \Lambda (\sigma)u \otimes v_{\sigma^{-1}(1)}\otimes \cdots \otimes v_{\sigma^{-1}(m)}
\]
defined by \(G\) and \(\Lambda\). The image of \(U\otimes V^{\otimes m}\) under the map \(S_\Lambda\) is called the generalized symmetry class of tensors associated with \(G\) and \(\Lambda\) and is denoted by \(V_\Lambda (G)\). The elements in \(V_\Lambda (G)\) of the form \(S_\Lambda (u\otimes v^\otimes)\) are called generalized decomposable tensors and are denoted by \(u{\circledast} v^{\circledast}\). For any linear operator \(T\) acting on \(V\), there is a unique induced operator \(K_\Lambda (T)\) acting on \(V_\Lambda (G)\) satisfying
\[
K_\Lambda (T)(u \otimes v^\otimes)=u\circledast Tv_1 \circledast \cdots \circledast Tv_m.
\]
If \(\dim U=1\), then \(K_\Lambda (T)\) reduces to \(K_\lambda (T)\), induced operator on symmetry class of tensors \(V_\lambda (G)\). In this paper, the basic properties of the induced operator \(K_\Lambda (T)\) are studied. Also some well-known results on the classical Schur functions will be extended to the case of generalized Schur functions.Green functions and Glauberman degree-divisibilityhttps://zbmath.org/1491.200262022-09-13T20:28:31.338867Z"Geck, Meinolf"https://zbmath.org/authors/?q=ai:geck.meinolfLet \(\Gamma \) be a finite group and \(S\) be a finite solvable group acting as a group of automorphisms on \(\Gamma \) where \(S\) and \(\Gamma \) have coprime orders. \textit{G. Glauberman} [Can. J. Math. 20, 1465--1488 (1968; Zbl 0167.02602)] introduced a bijection \(\theta \longmapsto \theta ^{\ast }\) between the set Irr\(_{S}(\Gamma )\) of \(S\)-invariant irreducible characters of \(\Gamma \) and the set Irr\((C_{\Gamma }(S))\) of irreducible characters of the centralizer of \(S\) in \(\Gamma \). The Glauberman degree-divisibility conjecture states that \(\theta ^{\ast }(1)\) always divides \(\theta (1)\ \)and this conjecture has been proved in special cases. In particular, \textit{B. Hartley} and \textit{A. Turull} [J. Reine Angew. Math. 451, 175--219 (1994; Zbl 0797.20007)] proved that the conjecture holds if \( \Gamma \) is a group of Lie type and a specific set of congruence conditions on Green functions on \(\Gamma \) are satisfied. They used this to prove the conjecture for some general classes of simple linear groups. In the present paper the author shows how Hartley and Turull's approach [loc. cit.] via congruence conditions on Green functions can be used to prove Glauberman's long-standing conjecture [loc. cit.] in complete generality. The proof pulls together extensive work on character sheaves by \textit{G. Lusztig} [Ann. Math. (2) 131, No. 2, 355--408 (1990; Zbl 0695.20024)] and \textit{T. Shoji} [Adv. Math. 111, No. 2, 244--313, 314--354 (1995; Zbl 0832.20065)].
Reviewer: John D. Dixon (Ottawa)One-dimensional pseudorepresentations with small defect that are trivial on a normal subgrouphttps://zbmath.org/1491.200272022-09-13T20:28:31.338867Z"Shtern, A. I."https://zbmath.org/authors/?q=ai:shtern.alexander-iSummary: We prove that every one-dimensional pseudorepresentation of a group with a sufficiently small defect that is trivial on a normal subgroup is defined by a one-dimensional pseudorepresentation of the corresponding quotient group.Geometrical equivalence and action type geometrical equivalence of group representationshttps://zbmath.org/1491.200282022-09-13T20:28:31.338867Z"Simões da Silva, J."https://zbmath.org/authors/?q=ai:simoes-da-silva.j"Tsurkov, A."https://zbmath.org/authors/?q=ai:tsurkov.arkadySummary: In this paper we construct an example of two representations \((V_1, G_1)\) and \((V_2, G_2)\) which are action type geometrically equivalent and groups \(G_1\) and \(G_2\) are geometrically equivalent, but the representations \((V_1, G_1)\) and \((V_2, G_2)\) are not geometrically equivalent.Some results on the main supergraph of finite groupshttps://zbmath.org/1491.200292022-09-13T20:28:31.338867Z"Asboei, A. K."https://zbmath.org/authors/?q=ai:khalili-asboei.alireza"Salehi, S. S."https://zbmath.org/authors/?q=ai:salehi.s-sSummary: Let \(G\) be a finite group. The main supergraph \(\mathcal{S}(G)\) is a graph with vertex set \(G\) in which two vertices \(x\) and \(y\) are adjacent if and only if \(o(x)|o(y)\) or \(o(y)| o(x)\). In this paper, we will show that \(G\cong \mathrm{PSL}(2, p)\) or \(\mathrm{PGL}(2, p)\) if and only if \(\mathcal{S}(G)\cong \mathcal{S}(\mathrm{PSL}(2, p))\) or \(\mathcal{S}(\mathrm{PGL}(2, p))\), respectively. Also, we will show that if \(M\) is a sporadic simple group, then \(G\cong M\) if only if \(\mathcal{S}(G)\cong \mathcal{S}(M)\).A new characterization of projective special linear groups \(L_3(q)\)https://zbmath.org/1491.200302022-09-13T20:28:31.338867Z"Ebrahimzadeh, B."https://zbmath.org/authors/?q=ai:ebrahimzadeh.behnamSummary: In this paper, we prove that projective special linear groups \(L_3(q)\), where \(0<q=5k\pm 2\) (\(k\in\mathbb{Z}\)) and \(q^2+q+1\) is a prime number can be uniquely determined by their order and the number of elements with same order.On residuals of finite groupshttps://zbmath.org/1491.200312022-09-13T20:28:31.338867Z"Aivazidis, Stefanos"https://zbmath.org/authors/?q=ai:aivazidis.stefanos"Müller, Thomas"https://zbmath.org/authors/?q=ai:muller.thomas-wInspired by \textit{S. Dolfi} et al. [Groups Geom. Dyn. 1, No. 4, 401--407 (2007; Zbl 1138.20018)], the authors of the paper under review extend the result to obtain the following interesting result:
If \(\mathcal{X}\) is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and \(\overline{\mathcal{X}}\) is the extension-closure of \(\mathcal{X}\), then there exists an constant \(\gamma\) such that, for all non-trivial finite groups \(G\) with trivial \(\mathcal{X}\)-radical,
\[
|G^{\overline{\mathcal{X}}}| > |G|^\gamma,
\]
where \(G^{\overline{\mathcal{X}}}\) is the \(\overline{\mathcal{X}}\)-residual of \(G\).
Reviewer: Yangming Li (Guangzhou)On products of generalised supersoluble finite groupshttps://zbmath.org/1491.200322022-09-13T20:28:31.338867Z"Ballester-Bolinches, A."https://zbmath.org/authors/?q=ai:ballester-bolinches.adolfo"Fakieh, W. M."https://zbmath.org/authors/?q=ai:fakieh.wafa-m|fakieh.wafaa-mohammed"Pedraza-Aguilera, M. C."https://zbmath.org/authors/?q=ai:pedraza-aguilera.mari-carmenSummary: In this paper, mutually sn-permutable subgroups of groups belonging to a class of generalised supersoluble groups are studied. Some analogs of known theorems on mutually sn-permutable products are established.On the exponent of the weak commutativity group \(\chi (G)\)https://zbmath.org/1491.200332022-09-13T20:28:31.338867Z"Bastos, R."https://zbmath.org/authors/?q=ai:bastos.raimundo"de Melo, E."https://zbmath.org/authors/?q=ai:de-melo.emerson"de Oliveira, R."https://zbmath.org/authors/?q=ai:de-oliveira.ronaldo-junio|de-oliveira.renata-z-g|de-oliveira.rafael-mendes|de-oliveira.roberson-assis|de-oliveira.ricardo-puziol|de-oliveira.r-i-jun|de-oliveira.rogerio-luiz-quintino-jun|de-oliveira.rosevaldo|de-oliveira.rodrigo-m-s|de-oliveira.rudinei-martins|de-oliveira.ricardo-t-a|de-oliveira.roberto-celio-limao|de-oliveira.rodolfo-alves|de-oliveira.rafael-massambone|de-oliveira.roberto-t-gSummary: The weak commutativity group \(\chi (G)\) is generated by two isomorphic groups \(G\) and \(G^{\varphi}\) subject to the relations \([g,g^{\varphi}]=1\) for all \(g \in G\). The group \(\chi (G)\) is an extension of \(D(G) = [G,G^{\varphi}]\) by \(G \times G\). We prove that if \(G\) is a finite solvable group of derived length \(d\), then \(\exp (D(G))\) divides \(\exp (G)^d\) if \(|G|\) is odd and \(\exp (D(G))\) divides \(2^{d-1}\cdot \exp (G)^d\) if \(|G|\) is even. Further, if \(p\) is a prime and \(G\) is a \(p\)-group of class \(p-1\), then \(\exp (D(G))\) divides \(\exp (G)\). Moreover, if \(G\) is a finite \(p\)-group of class \(c\geq 2\), then \(\exp (D(G))\) divides \(\exp (G)^{\lceil \log_{p-1}(c+1)\rceil} (p\geq 3)\) and \(\exp (D(G))\) divides \(2^{\lfloor \log_2(c)\rfloor} \cdot \exp (G)^{\lfloor \log_2(c)\rfloor +1} (p=2)\).Finite groups having nonnormal T.I. subgroupshttps://zbmath.org/1491.200342022-09-13T20:28:31.338867Z"Kızmaz, M. Yasir"https://zbmath.org/authors/?q=ai:yasir-kizmaz.m|kizmaz.muhammet-yasirA note on class sizes of vanishing elements in finite groupshttps://zbmath.org/1491.200352022-09-13T20:28:31.338867Z"Kong, Qingjun"https://zbmath.org/authors/?q=ai:kong.qingjun"Chen, Shi"https://zbmath.org/authors/?q=ai:chen.shiA generalization of subnormalityhttps://zbmath.org/1491.200362022-09-13T20:28:31.338867Z"Liu, A-Ming"https://zbmath.org/authors/?q=ai:liu.aiming|liu.aming"Guo, Wenbin"https://zbmath.org/authors/?q=ai:guo.wenbin.1"Safonova, Inna N."https://zbmath.org/authors/?q=ai:safonova.inna-n"Skiba, Alexander N."https://zbmath.org/authors/?q=ai:skiba.alexander-nSummary: Let \(G\) be a finite group and \(\mathfrak{F}\) a class of groups. A subgroup \(A\) of \(G\) is said to be \(\mathfrak{F}\)-\textit{normal} in \(G\) if either \(A\trianglelefteq G\) or \(A_G\ne A^G\) and every chief factor \(H/K\) of \(G\) between \(A_G\) and \(A^G\) is \(\mathfrak{F}\)-\textit{central} in \(G\), that is, \((H/K)\rtimes (G/C_G(H/K)) \in\mathfrak{F}\). We say that a subgroup \(H\) of \(G\) is \(\mathfrak{F}\wedge sn\text{-}embedded\) if for some \(\mathfrak{F}\)-normal subgroup \(A\) and subnormal subgroup \(B\) of \(G\) we have \(H=A\cap B\). In this paper, we analyze the influence of \(\mathfrak{F}\wedge sn\)-embedded subgroups on the structure of a group. In particular, we prove the following generalization of Schenkman's Theorem on the nilpotent residual of subnormal subgroups: If \(\mathfrak{F}\) is a hereditary saturated formation containing all nilpotent groups and \(S\) is an \(\mathfrak{F}\wedge sn\)-embedded subgroup of \(G\), such that \(Z_{\mathfrak{F}}(E)=1\) for every subgroup \(E\) of \(G\) containing \(S\), then \(C_G(S^{\mathfrak{F}})\le S^{\mathfrak{F}}\).Influence of complemented subgroups on the structure of finite groupshttps://zbmath.org/1491.200372022-09-13T20:28:31.338867Z"Malinowska, Izabela Agata"https://zbmath.org/authors/?q=ai:malinowska.izabela-agataSummary: P. Hall proved that a finite group \(G\) is supersoluble with elementary abelian Sylow subgroups if and only if every subgroup of \(G\) is complemented in \(G\). He called such groups complemented. \textit{A. Ballester-Bolinches} and \textit{X. Guo} [Arch. Math. 72, No. 3, 161--166 (1999; Zbl 0929.20015)] established the structure of minimal non-complemented groups. We give the classification of finite non-soluble groups all of whose second maximal subgroups are complemented groups. We also prove that every finite group with less than \(21\) non-complemented non-minimal \(\{2,3,5\}\)-subgroups is soluble.On products of finite w-supersoluble groupshttps://zbmath.org/1491.200382022-09-13T20:28:31.338867Z"Monakhov, Victor S."https://zbmath.org/authors/?q=ai:monakhov.victor-stepanovich"Trofimuk, Alexander A."https://zbmath.org/authors/?q=ai:trofimuk.aleksandr-aleksandrovichSummary: Let \(G\) be a finite group. A subgroup \(H\) of \(G\) is \(\mathbb{P}\)-\textit{subnormal} in \(G\) whenever either \(H=G\) or there exists a chain of subgroups \(H=H_0\leq H_1\leq \cdots \leq H_n=G\), such that \(|H_i:H_{i-1}|\) is a prime for every \(i=1, \dots,n\); \(G\) is said to be w-\textit{supersoluble} if every Sylow subgroup of \(G\) is \(\mathbb{P}\)-subnormal in \(G\). We study conditions under which the group \(G=AB\), where \(A\) and \(B\) are \(\mathbb{P}\)-subnormal subgroups of \(G\), belongs to a subgroup-closed saturated formation containing all finite supersoluble groups and contained in the class of all w-supersoluble groups.On maximal subformations of \(n\)-multiple \(\Omega \)-foliated formations of finite groupshttps://zbmath.org/1491.200392022-09-13T20:28:31.338867Z"Sorokina, Marina Mikhaĭlovna"https://zbmath.org/authors/?q=ai:sorokina.marina-m"Maksakov, Serafim Pavlovich"https://zbmath.org/authors/?q=ai:maksakov.serafim-pavlovichSummary: Only finite groups are considered in the article. Among the classes of groups the central place is occupied by classes closed regarding homomorphic images and subdirect products which are called formations. We study \(\Omega \)-foliateded formations constructed by V. A. Vedernikov in 1999 where \(\Omega\) is a nonempty subclass of the class \(\mathfrak{I}\) of all simple groups. \( \Omega \)-foliated formations are defined by two functions -- an \(\Omega \)-satellite \(f: \Omega \cup \{\Omega '\} \rightarrow \{ \text{formations}\}\) and a direction \(\varphi: \mathfrak{I} \rightarrow \{ \text{nonempty Fitting formations}\}\). The conception of multiple locality introduced by \textit{A. N. Skiba} in 1987 [Vopr. Algebry 3, 21--31 (1987; Zbl 0689.20015)] for formations and further developed for many other classes of groups, as applied to \(\Omega \)-foliated formations is as follows: every formation is considered to be 0-multiple \(\Omega \)-foliated with a direction \(\varphi \); an \(\Omega \)-foliated formation with a direction \(\varphi\) is called an \(n\)-multiple \(\Omega \)-foliated formation where \(n\) is a positive integer if it has such an \(\Omega \)-satellite all nonempty values of which are \((n-1)\)-multiple \(\Omega \)-foliated formations with the direction \(\varphi \). The aim of this work is to study the properties of maximal \(n\)-multiple \(\Omega \)-foliated subformations of a given \(n\)-multiple \(\Omega \)-foliated formation. We use classical methods of the theory of groups, of the theory of classes of groups, as well as methods of the general theory of lattices. In the paper we have established the existence of maximal \(n\)-multiple \(\Omega \)-foliated subformations for the formations with certain properties, we have obtained the characterization of the formation \(\Phi_{n\Omega\varphi} (\mathfrak{F})\) which is the intersection of all maximal \(n\)-multiple \(\Omega \)-foliated subformations of the formation \(\mathfrak{F}\), and we have revealed the relation between a maximal inner \(\Omega \)-satellite of 1-multiple \(\Omega \)-foliated formation and a maximal inner \(\Omega \)-satellite of its maximal 1-multiple \(\Omega \)-foliated subformation. The results will be useful in studying the inner structure of formations of finite groups, in particular, in studying the maximal chains of subformations and in establishing the lattice properties of formations.Characterization of finite groups with a unique non-nilpotent proper subgrouphttps://zbmath.org/1491.200402022-09-13T20:28:31.338867Z"Taeri, Bijan"https://zbmath.org/authors/?q=ai:taeri.bijan"Tayanloo-Beyg, Fatemeh"https://zbmath.org/authors/?q=ai:tayanloo-beyg.fatemehSummary: We characterize finite non-nilpotent groups \(G\) with a unique non-nilpotent proper subgroup. We show that \(|G|\) has at most three prime divisors. When \(G\) is supersolvable we find the presentation of \(G\) and when \(G\) is non-supersolvable we show that either \(G\) is a direct product of an Schmidt group and a cyclic group or a semi direct product of a \(p\)-group by a cyclic group of prime power order.Finite groups with generalized subnormal and generalized permutable subgroupshttps://zbmath.org/1491.200412022-09-13T20:28:31.338867Z"Zakrevskaya, Viktoria S."https://zbmath.org/authors/?q=ai:zakrevskaya.viktoria-sHilbert's theorem 90 for finite nilpotent groupshttps://zbmath.org/1491.200422022-09-13T20:28:31.338867Z"Cocke, William"https://zbmath.org/authors/?q=ai:cocke.william-jSummary: In this note we prove an analog of Hilbert's theorem 90 for finite nilpotent groups. Our version of Hilbert's theorem 90 was inspired by the Boston-Bush-Hajir (BBH) heuristics in number theory and will be useful in extending the BBH heuristics beyond quadratic field extensions.On the non-abelian tensor square of all groups of order dividing \(p^5\)https://zbmath.org/1491.200432022-09-13T20:28:31.338867Z"Ghorbanzadeh, Taleea Jalaeeyan"https://zbmath.org/authors/?q=ai:ghorbanzadeh.taleea-jalaeeyan"Parvizi, Mohsen"https://zbmath.org/authors/?q=ai:parvizi.mohsen"Niroomand, Peyman"https://zbmath.org/authors/?q=ai:niroomand.peymanAn extension of the Glauberman ZJ-theoremhttps://zbmath.org/1491.200442022-09-13T20:28:31.338867Z"Yasir Kızmaz, M."https://zbmath.org/authors/?q=ai:yasir-kizmaz.mIndices of maximal invariant subgroups and solvability of finite groupshttps://zbmath.org/1491.200452022-09-13T20:28:31.338867Z"Shao, Changguo"https://zbmath.org/authors/?q=ai:shao.changguo"Beltrán, Antonio"https://zbmath.org/authors/?q=ai:beltran.antonioSummary: Let \(A\) and \(G\) be finite groups and suppose that \(A\) acts coprimely on \(G\) via automorphisms. We study the solvability and supersolvability of \(G\) when certain proper maximal \(A\)-invariant subgroups of \(G\) have prime index or when they have certain prime power indices in \(G\).Schur's exponent conjecture -- counterexamples of exponent 5 and exponent 9https://zbmath.org/1491.200462022-09-13T20:28:31.338867Z"Vaughan-Lee, Michael"https://zbmath.org/authors/?q=ai:vaughan-lee.michael-r(no abstract)On the normal subgroup with minimal \(G\)-conjugacy class sizeshttps://zbmath.org/1491.200472022-09-13T20:28:31.338867Z"Zhao, Xianhe"https://zbmath.org/authors/?q=ai:zhao.xianhe"Zhou, Yanyan"https://zbmath.org/authors/?q=ai:zhou.yanyan"Chen, Ruifang"https://zbmath.org/authors/?q=ai:chen.ruifang"Zuo, Hongliang"https://zbmath.org/authors/?q=ai:zuo.hongliang"Huang, Qin"https://zbmath.org/authors/?q=ai:huang.qinSummary: Let \(N\) be a normal subgroup of a finite group \(G\), and \(x\) an element of \(N\). Objective that \(|x^G|=|G:C_G(x)|\), so \(|x^G|\) is called \textit{``minimal''} when \(C_G(x)\) is a maximal subgroup of \(G\). In this paper, we characterize the structure of \(N\) when \(|x^G|\) is minimal for every non-\(G\)-central element \(x\) of \(N\).Generalization of the basis property on finite groupshttps://zbmath.org/1491.200482022-09-13T20:28:31.338867Z"Al-Dayel, Ibrahim"https://zbmath.org/authors/?q=ai:al-dayel.ibrahim"Al Khalaf, Ahmad"https://zbmath.org/authors/?q=ai:al-khalaf.ahmadOn mutually \(m\)-permutable product of \(GS\)-groupshttps://zbmath.org/1491.200492022-09-13T20:28:31.338867Z"Abd-Ellatif, M. H."https://zbmath.org/authors/?q=ai:abd-ellatif.mohamed-hussein-hafez"Elkholy, A. M."https://zbmath.org/authors/?q=ai:elkholy.a-mFinite groups with given nearly \(S\Phi\)-embedded subgroupshttps://zbmath.org/1491.200502022-09-13T20:28:31.338867Z"Amjid, Venus"https://zbmath.org/authors/?q=ai:amjid.venus"Cao, Chenchen"https://zbmath.org/authors/?q=ai:cao.chenchen"Mao, Yuemei"https://zbmath.org/authors/?q=ai:mao.yuemeiOn \(\sigma\)-semipermutable subgroups of finite groupshttps://zbmath.org/1491.200512022-09-13T20:28:31.338867Z"Guo, Wen Bin"https://zbmath.org/authors/?q=ai:guo.wenbin.1"Skiba, Alexander N."https://zbmath.org/authors/?q=ai:skiba.alexander-nSummary: Let \(\sigma = \{\sigma_i\mid i \in I \}\) be some partition of the set of all primes \(\mathbb{P}\), \(G\) a finite group and \(\sigma(G) = \{\sigma_i|\sigma_i \cap \pi(G) \neq \emptyset\}\). A set \(\mathcal H\) of subgroups of \(G\) is said to be a \textit{complete Hall \(\sigma\)-set} of \(G\) if every member \(\neq\) 1 of \(\mathcal H\) is a Hall \(\sigma_i\)-subgroup of \(G\) for some \(\sigma_i \in \sigma\) and \(\mathcal H\) contains exactly one Hall \(\sigma_i\)-subgroup of \(G\) for every \(\sigma_i \in \sigma(G)\). A subgroup \(H\) of \(G\) is said to be: \(\sigma\)-\textit{semipermutable in} \(G\) \textit{with respect to} \(\mathcal H\) if \(HH_i ^x = H_i ^x H\) for all \(x \in G\) and all \(H_i\in \mathcal H\) such that \((|H|, |H_i|) = 1\); \(\sigma\)-\textit{semipermutable in} \(G\) if \(H\) is \(\sigma\)-semipermutable in \(G\) with respect to some complete Hall \(\sigma\)-set of \(G\). We study the structure of \(G\) being based on the assumption that some subgroups of \(G\) are \(\sigma\)-semipermutable in \(G\).The circular division of the factor group \(cf(Q_{2r}\times C_7,Z)/\overline{\mathcal{P}} (Q_{2r}\times C_7)\) when \(r\) is an odd numberhttps://zbmath.org/1491.200522022-09-13T20:28:31.338867Z"Jabir, Naba Hasoon"https://zbmath.org/authors/?q=ai:jabir.naba-hasoon"Abass, Rajaa Hassan"https://zbmath.org/authors/?q=ai:abass.rajaa-hassanSummary: The benefit of this paper is to find the circular divison of the factor group \(cf(Q_{2r}\times C_7,Z)/\overline{\mathcal{P}} (Q_{2r}\times C_7)\) when \(r\) is an odd number, where \(Q_{2r}\) is denoted to Quaternion group of order \(4r\), such that for each positive integer \(r\), there are two generators \(a\) and \(b\) for \(Q_{2r}\) satisfies \(Q_{2r} = \{a^h b^k, 0 \leq h \leq 2r - 1, k = 0, 1\}\) which has the following properties \(\{a^{2r} = b^4 = I, ba^rb^{-1} = a^{- r} \}\).Weakly totally permutable products and Fitting classeshttps://zbmath.org/1491.200532022-09-13T20:28:31.338867Z"Madanha, Sesuai Yash"https://zbmath.org/authors/?q=ai:madanha.sesuai-yashSummary: It is known that if \(G=AB\) is a product of its totally permutable subgroups \(A\) and \(B \), then \(G\in \mathfrak{F}\) if and only if \(A\in \mathfrak{F}\) and \(B\in \mathfrak{F}\) when \(\mathfrak{F}\) is a Fischer class containing the class \(\mathfrak{U}\) of supersoluble groups. We show that this holds when \(G=AB\) is a weakly totally permutable product for a particular Fischer class, \(\mathfrak{F}\diamond\mathfrak{N}\), where \(\mathfrak{F}\) is a Fitting class containing the class \(\mathfrak{U}\) and \(\mathfrak{N}\) a class of nilpotent groups. We also extend some results concerning the \(\mathfrak{U}\)-hypercentre of a totally permutable product to a weakly totally permutable product.On groups with conjugate-permutable subgroupshttps://zbmath.org/1491.200542022-09-13T20:28:31.338867Z"Murashka, Viachaslau I."https://zbmath.org/authors/?q=ai:murashka.viachaslau-iOn a product of two formational tcc-subgroupshttps://zbmath.org/1491.200552022-09-13T20:28:31.338867Z"Trofimuk, A."https://zbmath.org/authors/?q=ai:trofimuk.aleksandr-aleksandrovichSummary: A subgroup \(A\) of a group \(G\) is called tcc-\textit{subgroup} in \(G\), if there is a subgroup \(T\) of \(G\) such that \(G=AT\) and for any \(X\leqslant A\) and \(Y\leqslant T\) there exists an element \(u\in \langle X, Y\rangle\) such that \(XY^u\leqslant G\). The notation \(H\leqslant G\) means that \(H\) is a subgroup of a group \(G\). In this paper we consider a group \(G=AB\) such that \(A\) and \(B\) are tcc-subgroups in \(G\). We prove that \(G\) belongs to \(\mathfrak{F}\), when \(A\) and \(B\) belong to \(\mathfrak{F}\) and \(\mathfrak{F}\) is a saturated formation of soluble groups such that \(\mathfrak{U} \subseteq \mathfrak{F}\). Here \(\mathfrak{U}\) is the formation of all supersoluble groups.The probability of an automorphism of an abelian group fixing a group elementhttps://zbmath.org/1491.200562022-09-13T20:28:31.338867Z"Arora, Harsha"https://zbmath.org/authors/?q=ai:arora.harshaSummary: In this paper, we consider the probability of an automorphism of a finite abelian group fixing a group element. Explicit computations are made to find the fusion classes of finite abelian groups. The probability of an automorphism fixing a group element is obtained in terms of fusion classes. We also compute the bounds of the probability for some particular cases.OD-characterization of some simple unitary groupshttps://zbmath.org/1491.200572022-09-13T20:28:31.338867Z"Akbari, Majid"https://zbmath.org/authors/?q=ai:akbari.majid"Chen, Xiaoyou"https://zbmath.org/authors/?q=ai:chen.xiaoyou"Moghaddamfar, Alireza"https://zbmath.org/authors/?q=ai:moghaddamfar.ali-rezaSummary: We show that the simple unitary groups \(U_3(q)\) and \(U_4(q)\), with \(2< q < 10^2\) a prime power, are characterized among all groups by their order and the degrees of the vertices of their prime graph.Groups containing locally maximal product-free sets of size 4https://zbmath.org/1491.200582022-09-13T20:28:31.338867Z"Anabanti, C. S."https://zbmath.org/authors/?q=ai:anabanti.chimere-stanleySummary: Every locally maximal product-free set \(S\) in a finite group \(G\) satisfies \(G=S\cup SS \cup S^{-1}S \cup SS^{-1}\cup \sqrt{S}\), where \(SS=\{xy\mid x, y\in S\}\), \(S^{-1}S=\{x^{-1}y\mid x, y\in S\}\), \(SS^{-1}=\{xy^{-1}\mid x, y\in S\}\) and \(\sqrt{S}=\{x\in G\mid x^2\in S\}\). To better understand locally maximal product-free sets, \textit{E. A. Bertram} [Discrete Math. 44, 31--43 (1983; Zbl 0506.05060)] asked whether every locally maximal product-free set \(S\) in a finite abelian group satisfy \(|\sqrt{S}|\leqslant 2|S|\). This question was recently answered in the negation by the current author. Here, we improve some results on the structures and sizes of finite groups in terms of their locally maximal product-free sets. A consequence of our results is the classification of abelian groups that contain locally maximal product-free sets of size \(4\), continuing the work of \textit{A. P. Street} and \textit{E. G. Whitehead jun.} [J. Comb. Theory, Ser. A 17, 219--226 (1974; Zbl 0288.05020); Lect. Notes Math. 403, 109--124 (1974; Zbl 0318.05010)] and \textit{M. Giudici} and \textit{S. Hart} [Electron. J. Comb. 16, No. 1, Research Paper R59, 17 p. (2009; Zbl 1168.20009)] on the classification of groups containing locally maximal product-free sets of small sizes. We also obtain partial results on arbitrary groups containing locally maximal product-free sets of size \(4\), and conclude with a conjecture on the size \(4\) problem as well as an open problem on the general case.Some results on the join graph of finite groupshttps://zbmath.org/1491.200592022-09-13T20:28:31.338867Z"Bahrami, Zahara"https://zbmath.org/authors/?q=ai:bahrami.zahara"Taeri, Bijan"https://zbmath.org/authors/?q=ai:taeri.bijanSummary: Let \(G\) be a finite group which is not cyclic of prime power order. The join graph \(\Delta (G)\) of \(G\) is a graph whose vertex set is the set of all proper subgroups of \(G\), which are not contained in the Frattini subgroup \(G\) and two distinct vertices \(H\) and \(K\) are adjacent if and only if \(G=\langle H, K\rangle\). Among other results, we show that if \(G\) is a finite cyclic group and \(H\) is a finite group such that \(\Delta (G)\cong \Delta(H)\), then \(H\) is cyclic. Also we prove that \(\Delta (G) \cong\Delta (A_5)\) if and only if \(G\cong A_5\).On the maximal number of elements pairwise generating the symmetric group of even degreehttps://zbmath.org/1491.200602022-09-13T20:28:31.338867Z"Fumagalli, Francesco"https://zbmath.org/authors/?q=ai:fumagalli.francesco"Garonzi, Martino"https://zbmath.org/authors/?q=ai:garonzi.martino"Maróti, Attila"https://zbmath.org/authors/?q=ai:maroti.attilaIt is well known that the symmetric group \(S_n\) on \(n\) letters can be generated by two elements. Denote by \(\omega(S_n)\) the maximal size of a subset \(S \subseteq S_n\) such that \(\langle x, y \rangle = S_n\) for any distinct elements \(x,y\) of \(S\), and denote by \(\sigma(S_n)\) the minimal size of a set of proper subgroups of \(S_n\) whose set theoretic union is \(S_n\). It is not difficult to prove that \(\omega(S_n) \leqslant \sigma(S_n)\); however, exact values of these two invariants are extremely difficult to calculate in general. (Indeed, exact results for \(\sigma(S_n)\) are known only when \(n\) is odd or divisible by \(6\), and exact results for \(\omega(S_n)\) are known only when \(n\) is odd and at least \(17\).)
The purpose of this paper is to study the relationship between these two invariants when \(n\) is even. The authors prove that \(\omega(S_n)\) and \(\sigma(S_n)\) are both asymptotically \(\frac{1}{2}\binom{2n}{n}\) when \(n\) is even, and, in fact, \(\sigma(S_n)/\omega(S_n) \to 1\) as \(n \to \infty\). Furthermore, the authors prove related results about sets of elements of \(S_n\) that pairwise generate at least \(A_n\) and also provide CFSG-free lower bounds for \(\omega(S_n)\). The proofs are probabilistic in nature and rely on the Lovász local lemma.
Reviewer: Eric Swartz (Williamsburg)Characterization of groups with non-simple soclehttps://zbmath.org/1491.200612022-09-13T20:28:31.338867Z"Gorshkov, I. B."https://zbmath.org/authors/?q=ai:gorshkov.ilya-borisovichSummary: The spectrum of a finite group is the set of its element orders. We prove that if \(m>5\) then the group \(L_{2^m}(2)\times L_{2^m}(2)\times L_{2^m}(2)\) is uniquely determined by its spectrum in the class of finite groups.The minimum sum of element orders of finite groupshttps://zbmath.org/1491.200622022-09-13T20:28:31.338867Z"Jahani, Maghsoud"https://zbmath.org/authors/?q=ai:jahani.maghsoud"Marefat, Yadollah"https://zbmath.org/authors/?q=ai:marefat.yadollah"Refaghat, Hasan"https://zbmath.org/authors/?q=ai:refaghat.hasan"Vakili, Fasaghandisi Bahram"https://zbmath.org/authors/?q=ai:vakili.fasaghandisi-bahramSummary: Let \(G\) be a finite group and \(\psi (G)= \sum_{g\in G} o(g)\), where \(o(g)\) denotes the order of \(g\in G\). We show that the Conjecture 4.6.5 posed in [\textit{M. Herzog}, \textit{P. Longobardi} and \textit{M. Maj}, ``Properties of finite and periodic groups determined by their element orders (a survey)'', in: Group theory and computation. Singapore: Springer. 59--90 (2018; \url{doi:10.1007/978-981-13-2047-7_4})], is incorrect. In fact, we find a pair of finite groups \(G\) and \(S\) of the same order such that \(\psi (G)<\psi (S)\), with \(G\) solvable and \(S\) simple.A graph associated to centralizer of elements of a grouphttps://zbmath.org/1491.200632022-09-13T20:28:31.338867Z"Johari, Farangis"https://zbmath.org/authors/?q=ai:johari.farangis"Khashyarmanesh, Kazem"https://zbmath.org/authors/?q=ai:khashyarmanesh.kazemRecognition of 2-dimensional projective linear groups by the group order and the set of numbers of its elements of each orderhttps://zbmath.org/1491.200642022-09-13T20:28:31.338867Z"Khalili Asboei, Alireza"https://zbmath.org/authors/?q=ai:khalili-asboei.alirezaSummary: In a finite group \(G\), let \(\pi_{e}(G)\) be the set of orders of elements of \(G\), let \(s_{k}\) denote the number of elements of order \(k\) in \(G\), for each \(k\in \pi_{e}(G)\), and then let \(\mathrm{nse}(G)\) be the unordered set \(\{s_{k}:k\in{\pi}_{e}(G)\}\). In this paper, it is shown that if \(\vert G\vert=\vert L_{2}(q)\vert\) and \(\mathrm{nse}(G)=\mathrm{nse}(L_{2}(q))\) for some prime-power \(q\), then \(G\) is isomorphic to \(L_{2}(q)\).The number of subgroups of a given type in certain finite groupshttps://zbmath.org/1491.200652022-09-13T20:28:31.338867Z"Shelash, Hayder Baqer"https://zbmath.org/authors/?q=ai:shelash.hayder-baqer"Ashrafi, Ali Reza"https://zbmath.org/authors/?q=ai:ashrafi.alireza|ashrafi.ali-rezaSummary: The number of subgroups, normal subgroups and characteristic subgroups of a finite group \(G\) are denoted by \(Sub(G), NSub(G)\) and \(CSub(G)\), respectively. The main goal of this paper is to present a matrix model for computing these positive integers for dicyclic groups, semi-dihedral groups, and three sequences \(U_{6n}, V_{8n}\) and \(H(n)\) of groups that can be presented as follows:
\[
\begin{aligned}
U_{6n} &= \langle a, b \,|\, a^{2n} = b^3 = e, bab = a \rangle, \\
V_{8n} &= \langle a, b \,|\, a^{2n} = b^4 = e, aba = b^{-1}, ab^{-1}a = b \rangle, \\
H(n) &= \langle a,b,c \,|\, a^{2^{n-2}}=b^2=c^2=e, [a,b]=[b,c]=e, a^c=ab \rangle.
\end{aligned}
\] For each group, a matrix model containing all information is given.Arithmetic progressions in finite groupshttps://zbmath.org/1491.200662022-09-13T20:28:31.338867Z"Tărnăuceanu, Marius"https://zbmath.org/authors/?q=ai:tarnauceanu.mariusOn joins and intersections of subgroups in free groupshttps://zbmath.org/1491.200672022-09-13T20:28:31.338867Z"Ivanov, Sergei V."https://zbmath.org/authors/?q=ai:ivanov.sergei-vladimirovich.1Summary: We study graphs of (generalized) joins and intersections of finitely generated subgroups of a free group. We show how to disprove a lemma of Imrich and Müller on these graphs, how to repair the lemma and how to utilize it.Some properties of \(E(G, W, F_TG)\) and an application in the theory of splittings of groupshttps://zbmath.org/1491.200682022-09-13T20:28:31.338867Z"Fanti, E. L. C."https://zbmath.org/authors/?q=ai:fanti.erminia-de-lourdes-campello"Silva, L. S."https://zbmath.org/authors/?q=ai:silva.leticia-s|silva.lillia-s-b|silva.luciano-sSummary: Let us consider \(W\) a \(G\)-set and \(M\) a \(\mathbb{Z}_2G\)-module, where \(G\) is a group. In this paper we investigate some properties of the cohomological the theory of splittings of groups. Namely, we give a proof of the invariant \(E(G, W, M)\), defined in [\textit{M. G. C. Andrade} and \textit{E. de L. C. Fanti}, Int. J. Appl. Math. 25, No. 2, 183--190 (2012; Zbl 1282.20057)] and present related results with independence of \(E(G, W, M)\) with respect to the set of \(G\)-orbit representatives in \(W\) and properties of the invariant \(E(G, W, \mathcal{F}_TG)\) establishing a relation with the end of pairs of groups \(\widetilde{e}(G, T)\), defined by \textit{P. H. Kropholler} and \textit{M. A. Roller} [J. Pure Appl. Algebra 61, No. 2, 197--210 (1989; Zbl 0691.20036)]. The main results give necessary conditions for \(G\) to split over a subgroup \(T\), in the cases where \(M=\mathbb{Z}_2(G/T)\) or \(M=\mathcal{F}_TG\).On the structure of some groups having finite contranormal subgroupshttps://zbmath.org/1491.200692022-09-13T20:28:31.338867Z"Kurdachenko, L. A."https://zbmath.org/authors/?q=ai:kurdachenko.leonid-a"Semko, N. N."https://zbmath.org/authors/?q=ai:semko.nikolaj-nSummary: Following \textit{J. S. Rose} [Math. Z. 106, 97--112 (1968; Zbl 0169.03402)], a subgroup \(H\) of the group \(G\) is said to be contranormal in \(G\), if \(G=H^G\). In a certain sense, contranormal subgroups are antipodes to subnormal subgroups. We study the structure of Abelian-by-nilpotent groups having a finite proper contranormal \(p\)-subgroup.On growth of generalized Grigorchuk's overgroupshttps://zbmath.org/1491.200702022-09-13T20:28:31.338867Z"Samarakoon, S. T."https://zbmath.org/authors/?q=ai:samarakoon.supun-tSummary: Grigorchuk's Overgroup \(\widetilde{\mathcal{G}}\), is a branch group of intermediate growth. It contains the first Grigorchuk's torsion group \(\mathcal{G}\) of intermediate growth constructed in 1980, but also has elements of infinite order. Its growth is substantially greater than the growth of \(\mathcal{G}\). The group \(\mathcal{G}\), corresponding to the sequence \((012)^\infty = 012012 \dots\), is a member of the family \(\{G_\omega\mid \omega \in \Omega = \{ 0, 1, 2\}^\mathbb{N}\}\) consisting of groups of intermediate growth when sequence \(\omega\) is not eventually constant. Following this construction we define the family \(\{\widetilde{G}_\omega, \omega \in \Omega\}\) of generalized overgroups. Then \(\widetilde{\mathcal{G}} = \widetilde{G}_{(012)^\infty}\) and \(G_\omega\) is a subgroup of \(\widetilde{G}_\omega\) for each \(\omega \in \Omega\). We prove, if \(\omega\) is eventually constant, then \(\widetilde{G}_\omega\) is of polynomial growth and if \(\omega\) is not eventually constant, then \(\widetilde{G}_\omega\) is of intermediate growth.Groups satisfying chain conditions on \(f\)-subnormal subgroupshttps://zbmath.org/1491.200712022-09-13T20:28:31.338867Z"Dixon, Martyn R."https://zbmath.org/authors/?q=ai:dixon.martyn-russell"Ferrara, Maria"https://zbmath.org/authors/?q=ai:ferrara.maria"Trombetti, Marco"https://zbmath.org/authors/?q=ai:trombetti.marcoSummary: In this paper, we study groups with various chain conditions on \(f\)-subnormal subgroups. A subgroup \(H\) of a group \(G\) is called \(f\)-subnormal in \(G\) if there is a finite chain of subgroups \(H=H_0\leq H_1\leq \cdots \leq H_n=G \), such that either \(|H_{i+1}: H_i|\) is finite or \(H_i\) is normal in \(H_{i+1}\), for \(0\leq i\leq n-1\).Groups whose lattices of normal subgroups are factorialhttps://zbmath.org/1491.200722022-09-13T20:28:31.338867Z"Rajhi, A."https://zbmath.org/authors/?q=ai:rajhi.anisSummary: We prove that the groups \(G\) for which the lattice of normal subgroups \(\mathcal{N}(G)\) is factorial are exactly the UND-groups, that is the groups for which every normal subgroup have a unique normal complement, with finite length.Remarks on profinite groups having few open subgroupshttps://zbmath.org/1491.200732022-09-13T20:28:31.338867Z"Segal, Dan"https://zbmath.org/authors/?q=ai:segal.danSummary: We explore the relations between various conditions of ``smallness'' on profinite groups.Decomposition of groups and the wreath product of permutation groupshttps://zbmath.org/1491.200742022-09-13T20:28:31.338867Z"Ghadbane, Nacer"https://zbmath.org/authors/?q=ai:ghadbane.nacerSummary: In this work, we study decomposition of groups. Let \(G\) be a group, and let \(N\) be a normal subgroup of \(G\); we shall show how multiplication in \(G\) can be viewed as a two-step process consisting of multiplication in the quotient \(G/N\) followed by multiplication in \(N\). The object of wreath product of permutation groups is defined by the actions on Cartesian product of two sets. In this paper we consider \(S(\Gamma)\) and \(S(\Delta)\) -- the permutation groups on \(\Gamma\) and \(\Delta\) respectively, and \(S(\Gamma)^\Delta\) -- the set of all maps of \(\Delta\) into the permutations group \(S(\Gamma)\), to provide the wreath product \(W\) of \(S(\Gamma)\) by \(S(\Delta)\), and the action of \(W\) on \(\Gamma\times\Delta\).On infinite groups whose finite quotients have restricted prime divisorshttps://zbmath.org/1491.200752022-09-13T20:28:31.338867Z"Robinson, Derek J. S."https://zbmath.org/authors/?q=ai:robinson.derek-j-sSummary: The effect of restricting the set of primes dividing the orders of the finite quotients of a group is investigated. Particular attention is paid to abelian, soluble, locally soluble and locally finite groups. The connection with the extraction of roots is explored.Conditions of Dedekindness of generalized norms in nonperiodic groupshttps://zbmath.org/1491.200762022-09-13T20:28:31.338867Z"Lukashova, T. D."https://zbmath.org/authors/?q=ai:lukashova.tetyana-d"Drushlyak, M. G."https://zbmath.org/authors/?q=ai:drushlyak.maryna-g"Lyman, F. M."https://zbmath.org/authors/?q=ai:lyman.fedir-mykolaiovychThe Fibonacci-circulant sequences in the binary polyhedral groupshttps://zbmath.org/1491.200772022-09-13T20:28:31.338867Z"Karaduman, Erdal"https://zbmath.org/authors/?q=ai:karaduman.erdal"Deveci, Omur"https://zbmath.org/authors/?q=ai:deveci.omurSummary: In 2017 \textit{Ö. Deveci} et al. [Iran. J. Sci. Technol., Trans. A, Sci. 41, No. 4, 1033--1038 (2017; Zbl 1391.11026)] defined the Fibonacci-circulant sequences of the first and second kinds as shown, respectively:
\[
x_n^1 = -x_{n-1}^1+x_{n-2}^1-x_{n-3}^1\text{ for }n \geq 4,\text{ where } x_1^1=x_2^1=0\text{ and } x_3^1=1
\]
and
\[
x_n^2 = -x_{n-3}^2-x_{n-4}^2+x_{n-5}^2\text{ for }n \geq 6, \text{ where }x_1^2 = x_2^2 = x_3^2=x_4^2=0\text{ and }x_5^2=1
\]
Also, they extended the Fibonacci-circulant sequences of the first and second kinds to groups. In this paper, we obtain the periods of the Fibonacci-circulant sequences of the first and second kinds in the binary polyhedral groups.Knapsack problems for wreath productshttps://zbmath.org/1491.200782022-09-13T20:28:31.338867Z"Ganardi, Moses"https://zbmath.org/authors/?q=ai:ganardi.moses"König, Daniel"https://zbmath.org/authors/?q=ai:konig.daniel"Lohrey, Markus"https://zbmath.org/authors/?q=ai:lohrey.markus"Zetzsche, Georg"https://zbmath.org/authors/?q=ai:zetzsche.georgSummary: In recent years, knapsack problems for (in general non-commutative) groups have attracted attention. In this paper, the knapsack problem for wreath products is studied. It turns out that decidability of knapsack is not preserved under wreath product. On the other hand, the class of knapsack-semilinear groups, where solutions sets of knapsack equations are effectively semilinear, is closed under wreath product. As a consequence, we obtain the decidability of knapsack for free solvable groups. Finally, it is shown that for every non-trivial abelian group \(G\), knapsack (as well as the related subset sum problem) for the wreath product \(G\wr\mathbb{Z}\) is \textsf{NP}-complete.
For the entire collection see [Zbl 1381.68010].Groups whose word problems are not semilinearhttps://zbmath.org/1491.200792022-09-13T20:28:31.338867Z"Gilman, Robert H."https://zbmath.org/authors/?q=ai:gilman.robert-h"Kropholler, Robert P."https://zbmath.org/authors/?q=ai:kropholler.robert-p"Schleimer, Saul"https://zbmath.org/authors/?q=ai:schleimer.saulSummary: Suppose that \(G\) is a finitely generated group and \({\mathrm{WP}(G)}\) is the formal language of words defining the identity in \(G\). We prove that if \(G\) is a virtually nilpotent group that is not virtually abelian, the fundamental group of a finite volume hyperbolic three-manifold, or a right-angled Artin group whose graph lies in a certain infinite class, then \({\mathrm{WP}(G)}\) is not a multiple context-free language.Coloring invariants of knots and links are often intractablehttps://zbmath.org/1491.200802022-09-13T20:28:31.338867Z"Kuperberg, Greg"https://zbmath.org/authors/?q=ai:kuperberg.greg"Samperton, Eric"https://zbmath.org/authors/?q=ai:samperton.ericOne of the main problems in knot theory is the knot recognition problem. The essence of this problem is to construct an effective algorithm that determines by two given knots whether they are equivalent or not. One possible approach to this problem is to construct invariants, i.e. such functions that have the same value on equivalent knots. If we manage to find an invariant \(f\) such that \(f(K_1)\neq f(K_2)\), then we can conclude that \(K_1\) and \(K_2\) are non-equivalent knots.
One of the simplest invariants in knot theory is the \(3\)-coloring invariant of the diagram which is also called the Fox coloring. This invariant is not very strong, i.e. it can be the same for a lot of different knots. However, it allows to distinguish the trefoil knot from the trivial knot and from the figure-eight knot.
More complex invariants are the knot group and the knot quandle. These invariants are very strong (in particular, the knot quandle defines a knot up to simultaneous change of orientation and mirror reflection), however, in order to use these invariants one must be able to solve the isomorphism problem for groups or the isomorphism problem for quandles. In [\textit{A. D. Brooke-Taylor} and \textit{S. K. Miller}, J. Aust. Math. Soc. 108, No. 2, 262--277 (2020; Zbl 1482.20039)] it is shown that the isomorphism problem for quandles is, from the perspective of Borel reducibility, fundamentally difficult (Borel complete).
The invariant \(|Q(K;G,C)|\), where \(K\) is a knot, \(G\) is a nonabelian finite simple group, \(C\) is a generating conjugacy class in \(G\), is a kind of intermediate between the knot group and the knot coloring. The value of this invariant on a knot \(K\) is equal to the number of elements in the quotient of the set of surjective homomorphisms from the group \(G(K)\) of knot \(K\) to the group \(G\) such that the images of the Wirtinger generators of the group \(G(K)\) belong to \(C\), by the group \(\mathrm{Aut}(G,C)\) of automorphisms of \(G\) which fix \(C\).
The authors of the paper under review prove that the problem of computing the invariant \(|Q(K;G,C)|\) for a given group \(G\) and its conjugcay class \(C\) is parsimoniously \(\#P\)-complete.
Reviewer: Timur Nasybullov (Novosibirsk)On finitely generated submonoids of virtually free groupshttps://zbmath.org/1491.200812022-09-13T20:28:31.338867Z"Silva, Pedro V."https://zbmath.org/authors/?q=ai:silva.pedro-v"Zakharov, Alexander"https://zbmath.org/authors/?q=ai:zakharov.alexander-a|zakharov.alexander-fSummary: We prove that it is decidable whether or not a finitely generated submonoid of a virtually free group is graded, introduce a new geometric characterization of graded submonoids in virtually free groups as quasi-geodesic submonoids, and show that their word problem is rational (as a relation). We also solve the isomorphism problem for this class of monoids, generalizing earlier results for submonoids of free monoids. We also prove that the classes of graded monoids, regular monoids and Kleene monoids coincide for submonoids of free groups.Some properties of left \(2_\alpha\)-Engel subgroupshttps://zbmath.org/1491.200822022-09-13T20:28:31.338867Z"Mehrjerdi, Ali Mohammad Z."https://zbmath.org/authors/?q=ai:mehrjerdi.ali-mohammad-z"Moghaddam, Mohammad Reza R."https://zbmath.org/authors/?q=ai:moghaddam.mohammad-reza-r"Rostamyari, Mohammad Amin"https://zbmath.org/authors/?q=ai:rostamyari.mohammad-aminOn Baer's theorem and its generalizationshttps://zbmath.org/1491.200832022-09-13T20:28:31.338867Z"Taghavi, Yasaman"https://zbmath.org/authors/?q=ai:taghavi.yasaman"Kayvanfar, Saeed"https://zbmath.org/authors/?q=ai:kayvanfar.saeed"Parvizi, Mohsen"https://zbmath.org/authors/?q=ai:parvizi.mohsenSummary: A well known theorem of R. Baer states that if \(G\) is a group and \(G/Z_n(G)\) is finite, then \(\gamma_{n+1}(G)\) is finite. In this article, we extend this theorem for groups \(G\) that have subgroups \(A\) of \(\Aut(G)\) such that \(A/\operatorname{Inn}(G)\) is finitely generated or is of finite special rank. Furthermore, some new upper bounds for \(|\gamma_{n+1}(G)|\) and \(|\gamma_{n+1}(G,A)|\) are presented.On the structure of some locally nilpotent groups without contranormal subgroupshttps://zbmath.org/1491.200842022-09-13T20:28:31.338867Z"Kurdachenko, Leonid A."https://zbmath.org/authors/?q=ai:kurdachenko.leonid-a"Longobardi, Patrizia"https://zbmath.org/authors/?q=ai:longobardi.patrizia"Maj, Mercede"https://zbmath.org/authors/?q=ai:maj.mercedeA subgroup \(H\) of a group \(G\) is contranormal in \(G\) if \(H^G=G\), a term introduced by \textit{J. S. Rose} [Proc. Lond. Math. Soc. (3) 17, 447--469 (1967; Zbl 0153.03602); Math. Z. 106, 97--112 (1968; Zbl 0169.03402)]. Important examples of contranormal subgroups are abnormal subgroups. However, a locally nilpotent group contains no proper abnormal subgroups, whereas the locally dihedral \(2\)-group is a hypercentral group which contains a proper contranormal subgroup. It is easy to see that a finite group is nilpotent if and only if it has no proper contranormal subgroups. However in infinite groups there are non-nilpotent (even non locally nilpotent) examples with all proper subgroups subnormal and hence there are no proper contranormal subgroups in such a case. A natural question then is to determine what can be said concerning groups which have no proper contranormal subgroups. This question has been studied by various authors (see [\textit{L. A. Kurdachenko} et al., Commun. Algebra 37, No. 12, 4542--4557 (2009; Zbl 1192.20015); \textit{B. A. F. Wehrfritz}, Publ. Mat., Barc. 64, No. 1, 183--194 (2020; Zbl 1445.20030)], for example). In the current paper, this question is explored further. For example, the authors prove: Proposition A: Let \(G\), be a group, let \(L, K\) be normal subgroups of \(G\) such that \(L\) is a periodic locally nilpotent subgroup, \(L\leq K\) and \(K/L\) is finite. If \(G\) contains no proper contranormal subgroups, then \(K\) is locally nilpotent. This result itself has many interesting consequences. The authors go on to obtain more conditions on ``contranormal-free'' groups which imply nilpotency, as in the finite case. Overall this is a very interesting, informative paper with many useful techniques and results.
Reviewer: Martyn Dixon (Tuscaloosa)Locally finite \(p\)-groups with a left 3-Engel element whose normal closure is not nilpotenthttps://zbmath.org/1491.200852022-09-13T20:28:31.338867Z"Hadjievangelou, Anastasia"https://zbmath.org/authors/?q=ai:hadjievangelou.anastasia"Noce, Marialaura"https://zbmath.org/authors/?q=ai:noce.marialaura"Traustason, Gunnar"https://zbmath.org/authors/?q=ai:traustason.gunnarThe probability that a pair of group elements is 2-Engelhttps://zbmath.org/1491.200862022-09-13T20:28:31.338867Z"Khoshtarash, Raheleh"https://zbmath.org/authors/?q=ai:khoshtarash.raheleh"Moghaddam, Mohammad Reza R."https://zbmath.org/authors/?q=ai:moghaddam.mohammad-reza-r"Rostamyari, Mohammad Amin"https://zbmath.org/authors/?q=ai:rostamyari.mohammad-aminOn periodic groups saturated with finite Frobenius groupshttps://zbmath.org/1491.200872022-09-13T20:28:31.338867Z"Durakov, Boris E."https://zbmath.org/authors/?q=ai:durakov.boris-evgenevich"Sozutov, Anatoly I."https://zbmath.org/authors/?q=ai:sozutov.anatolii-ilichSummary: A group is called weakly conjugate biprimitively finite if each its element of prime order generates a finite subgroup with any of its conjugate elements. A binary finite group is a periodic group in which any two elements generate a finite subgroup. If \(\mathfrak{X}\) is some set of finite groups, then the group \(G\) saturated with groups from the set \(\mathfrak{X}\) if any finite subgroup of \(G\) is contained in a subgroup of \(G\), isomorphic to some group from \(\mathfrak{X} \). A group \(G = F \leftthreetimes H\) is a Frobenius group with kernel \(F\) and a complement \(H\) if \(H \cap H^f = 1\) for all \(f \in F^{\#}\) and each element from \(G \setminus F\) belongs to a one conjugated to \(H\) subgroup of \(G\). In the paper we prove that a saturated with finite Frobenius groups periodic weakly conjugate biprimitive finite group with a nontrivial locally finite radical is a Frobenius group. A number of properties of such groups and their quotient groups by a locally finite radical are found. A similar result was obtained for binary finite groups with the indicated conditions. Examples of periodic non locally finite groups with the properties above are given, and a number of questions on combinatorial group theory are raised.On two properties of Shunkov grouphttps://zbmath.org/1491.200882022-09-13T20:28:31.338867Z"Shlepkin, Alekseĭ Anatol'evich"https://zbmath.org/authors/?q=ai:shlepkin.aleksei-anatolevich"Sabodakh, Irina Valer'evna"https://zbmath.org/authors/?q=ai:sabodakh.irina-valerevnaSummary: One of the interesting classes of mixed groups (i.e. groups that can contain both elements of finite order and elements of infinite order) is the class of Shunkov groups. The group \(G\) is called Shunkov group if for any finite subgroup \(H\) of \(G\) in the quotient group \(N_G(H)/H\), any two conjugate elements of prime order generate a finite group. When studying the Shunkov group \(G\), a situation often arises when it is necessary to move to the quotient group of the group \(G\) by some of its normal subgroup \(N\). In which cases is the resulting quotient group \(G/N\) again a Shunkov group? The paper gives a positive answer to this question, provided that the normal subgroup \(N\) is locally finite and the orders of elements of the subgroup \(N\) are mutually simple with the orders of elements of the quotient group \(G/N\).
Let \(\mathfrak{X}\) be a set of groups. A group \(G\) is saturated with groups from the set \(\mathfrak{X}\) if any finite subgroup of \(G\) is contained in a subgroup of \(G\) that is isomorphic to some group of \(\mathfrak{X} \). If all elements of finite orders from the group \(G\) are contained in a periodic subgroup of the group \(G\), then it is called the periodic part of the group \(G\) and is denoted by \(T(G)\). It is proved that the Shunkov group saturated with finite linear and unitary groups of degree 3 over finite fields has a periodic part that is isomorphic to either a linear or unitary group of degree 3 on a suitable locally finite field.Proper actions on finite products of quasi-treeshttps://zbmath.org/1491.200892022-09-13T20:28:31.338867Z"Bestvina, Mladen"https://zbmath.org/authors/?q=ai:bestvina.mladen"Bromberg, Ken"https://zbmath.org/authors/?q=ai:bromberg.kenneth-w"Fujiwara, Koji"https://zbmath.org/authors/?q=ai:fujiwara.koji.1Say that a group \(G\) has property (QT) if it acts isometrically on a finite product of quasi-trees such that orbit maps are quasi-isometric embeddings. Here, a \textit{quasi-tree} is a connected graph whose path metric is quasi-isometric to a tree and the product is given the \(\ell^1\)-metric. In this paper, the authors prove that residually finite Gromov hyperbolic groups have (QT), as do mapping class groups of finite-type surfaces.
To build intuition, observe that it is clear that free groups have (QT), and by choosing a filling collection of simple closed curves on a surface and taking the Bass-Serre trees dual to that collection, it is not hard to see that surface groups have (QT); in each of these cases there is in fact a proper action on a finite product of \textit{trees,} a property shared by (subgroups of) Coxeter groups by a result of \textit{A. Dranishnikov} and \textit{T. Januszkiewicz} [Topol. Proc. 24(Spring), 135--141 (1999; Zbl 0973.20029). It follows that \emph{virtually cocompact special} cubulated groups have (QT), a class which includes fundamental groups of closed hyperbolic 3-manifolds. These latter groups act without global fixed point on CAT(0) cube complexes, and thus have the \emph{Haagerup property,} a strong negation of Kazhdan's property (T).
On the other hand, uniform lattices in \(\mathrm{Sp}(n,1)\) have Kazhdan's property (T) and are residually finite and Gromov hyperbolic. Therefore the authors' result shows that property (QT) is not strong enough to imply the Haagerup property. Indeed, it is unknown whether mapping class groups of finite-type surfaces have either the Haagerup property or Kazhdan's property (T).
Since finite products of (quasi-)trees have finite asymptotic dimension, property (QT) is a stronger version of finiteness of the asymptotic dimension, a result due to Gromov for Gromov hyperbolic groups and to the authors [Publ. Math., Inst. Hautes Étud. Sci. 122, 1--64 (2015; Zbl 1372.20029)] for mapping class groups of finite-type surfaces. Indeed, the proof uses the projection complexes techniques developed in the authors' previous paper. The authors remark that lattices in higher-rank Lie groups have finite asymptotic dimension but cannot have (QT). Thus in this regard the mapping class group of a finite-type surface behaves more like a Gromov hyperbolic group than a (higher-rank) lattice.
Reviewer: Rylee Lyman (Newark)Negative curvature in graph braid groupshttps://zbmath.org/1491.200902022-09-13T20:28:31.338867Z"Genevois, Anthony"https://zbmath.org/authors/?q=ai:genevois.anthonyA note on locally elliptic actions on cube complexeshttps://zbmath.org/1491.200912022-09-13T20:28:31.338867Z"Leder, Nils"https://zbmath.org/authors/?q=ai:leder.nils-jonas"Varghese, Olga"https://zbmath.org/authors/?q=ai:varghese.olgaSummary: We deduce from Sageev's results that whenever a group acts locally elliptically on a finite-dimensional \({\mathrm {CAT}}(0)\) cube complex, then it must fix a point. As an application, we partially prove a conjecture by Marquis concerning actions on buildings and we give an example of a group \(G\) such that \(G\) does not have property (T), but \(G\) and all its finitely generated subgroups can not act without a fixed point on a finite-dimensional \({\mathrm {CAT}}(0)\) cube complex, answering a question by Barnhill and Chatterji.On Jones' subgroup of R. Thompson's group \(T\)https://zbmath.org/1491.200922022-09-13T20:28:31.338867Z"Nikkel, Jordan"https://zbmath.org/authors/?q=ai:nikkel.jordan"Ren, Yunxiang"https://zbmath.org/authors/?q=ai:ren.yunxiangQuotient and blow-up of automorphisms of graphs of groupshttps://zbmath.org/1491.200932022-09-13T20:28:31.338867Z"Ye, Kaidi"https://zbmath.org/authors/?q=ai:ye.kaidiA new combination theorem for relatively hyperbolic groupshttps://zbmath.org/1491.200942022-09-13T20:28:31.338867Z"Kim, Jan"https://zbmath.org/authors/?q=ai:kim.yan|kim.jan-t"Lee, Donghi"https://zbmath.org/authors/?q=ai:lee.donghiSummary: We prove a new combination theorem for relatively hyperbolic groups by analyzing diagrams over HNN-extensions of relatively hyperbolic groups.Corrigendum to: ``Graphs of hyperbolic groups and a limit set intersection theorem''https://zbmath.org/1491.200952022-09-13T20:28:31.338867Z"Sardar, Pranab"https://zbmath.org/authors/?q=ai:sardar.pranabSummary: The purpose of this note is to point out a mistake in the proof of Proposition 4.9 of the author's paper [ibid. 146, No. 5, 1859--1871 (2018; Zbl 1486.20062)] and the consequences thereof.Characteristic and IA-invariant radicalshttps://zbmath.org/1491.200962022-09-13T20:28:31.338867Z"Samadi, Mohammad"https://zbmath.org/authors/?q=ai:samadi.mohammad-aref"Shahryari, Mohammad"https://zbmath.org/authors/?q=ai:shahryari.mohammad-reza-balooch|shahryari.mohammadSummary: In this article, we study systems of coefficient-free equations over special classes of groups, their radicals in which are characteristic or IA-invariant subgroups of the free group.The Yagita invariant of symplectic groups of large rankhttps://zbmath.org/1491.200972022-09-13T20:28:31.338867Z"Busch, Cornelia M."https://zbmath.org/authors/?q=ai:busch.cornelia-minette"Leary, Ian J."https://zbmath.org/authors/?q=ai:leary.ian-jamesSummary: Fix a prime \(p\), and let \(\mathcal{O}\) denote a subring of \(\mathbb{C}\) that is either integrally closed or contains a primitive \(p\)th root of 1. We determine the Yagita invariant at the prime \(p\) for the symplectic group \(\mathrm{Sp}(2n,\mathcal{O})\) for all \(n\ge p-1\).Presentations of general linear groups with Jordan regular generatorshttps://zbmath.org/1491.200982022-09-13T20:28:31.338867Z"Sahai, Meena"https://zbmath.org/authors/?q=ai:sahai.meena"Kumari, Parvesh"https://zbmath.org/authors/?q=ai:kumari.parvesh"Sharma, R. K."https://zbmath.org/authors/?q=ai:sharma.rahul-kumar|sharma.ravi-k|sharma.rajan-k|sharma.rajneesh-k|sharma.rajendra-kumar|sharma.ravinder-kumar|sharma.raj-kumar|sharma.ruchi-kaushik|sharma.rakesh-kumar|sharma.ram-kishore|sharma.ram-krishan|sharma.ram-krishnan|sharma.rishi-kumarSummary: In this article, we obtain presentations of the general linear groups \(\mathrm{GL}(2, \mathbb{Z}_{16})\), \(\mathrm{GL}(2, \mathbb{Z}_{18})\), \(\mathrm{GL}(2, \mathbb{Z}_{20})\), \(\mathrm{GL}(2, \mathbb{Z}_{24})\), \(\mathrm{GL}(2, \mathbb{Z}_{28})\), \(\mathrm{GL}(2, \mathbb{Z}_{30})\), \(\mathrm{GL}(2, \mathbb{Z}_{32})\), \(\mathrm{GL}(2, \mathbb{Z}_{36})\), \(\mathrm{GL}(2, \mathbb{Z}_{38})\), \(\mathrm{GL}(2, \mathbb{Z}_{40})\) and \(\mathrm{GL}(2, \mathbb{Z}_{42})\) with Jordan regular units as generators.Group-theoretical graph categorieshttps://zbmath.org/1491.200992022-09-13T20:28:31.338867Z"Gromada, Daniel"https://zbmath.org/authors/?q=ai:gromada.danielThe paper under review is concerned with certain diagrammatic categories of partitions and graphs and with their relations with compact matrix quantum groups and their representation categories, as considered in [\textit{S.L.\ Woronowicz}, Invent.\ Math.\ 93, No.\ 1, 35--76 (1988; Zbl 0664.58044)]. Categories of partitions were classified in [\textit{S. Raum} and \textit{M. Weber}, Commun. Math. Phys. 341, No. 3, 751--779 (2016; Zbl 1356.46061)], and an important class of such categories is formed by the group theoretical categories of partitions. The compact matrix quantum groups corresponding to group theoretical categories of partitions were described in [\textit{S.\ Raum} and \textit{M.\ Weber}, J.\ Noncommut.\ Geom.\ 9, No.\ 4, 1261--1293 (2015; Zbl 1350.46047)], and a more general class of skew categories of partitions was studied in [\textit{L. Maaßen}, J. Noncommut. Geom. 14, No. 3, 987--1017 (2020; Zbl 07358364)]. The present paper generalizes these concepts into the context of graph categories. The author defines a class of \emph{group theoretical graph categories} and introduces the more general concept of \emph{skew graph categories}. He also defines \emph{graph fibrations}, shows that they are in one-to-one correspondence with skew graph categories and uses them to describe the compact matrix quantum group corresponding to such a category. In the final section, the opposite question of describing graph categories corresponding to a given compact matrix quantum group is considered.
Reviewer: Jonathan Gruber (Lausanne)Boundedly finite conjugacy classes of tensorshttps://zbmath.org/1491.201002022-09-13T20:28:31.338867Z"Bastos, Raimundo"https://zbmath.org/authors/?q=ai:bastos.raimundo"Monetta, Carmine"https://zbmath.org/authors/?q=ai:monetta.carmineSummary: Let \(n\) be a positive integer and let \(G\) be a group. We denote by \(\nu (G)\) a certain extension of the non-abelian tensor square \(G \otimes G\) by \(G \times G\). Set \(T_\otimes (G) = \{g \otimes h \mid g,h \in G\}\). We prove that if the size of the conjugacy class \(\left |x^{\nu (G)} \right| \leq n\) for every \(x \in T_\otimes (G)\), then the second derived subgroup \(\nu (G)''\) is finite with \(n\)-bounded order. Moreover, we obtain a sufficient condition for a group to be a BFC-group.On non-split abelian extensions. IIhttps://zbmath.org/1491.201012022-09-13T20:28:31.338867Z"Snanou, Noureddine"https://zbmath.org/authors/?q=ai:snanou.noureddineOn non-split abelian extensionshttps://zbmath.org/1491.201022022-09-13T20:28:31.338867Z"Snanou, Noureddine"https://zbmath.org/authors/?q=ai:snanou.noureddine"Charkani, Mohammed Elhassani"https://zbmath.org/authors/?q=ai:charkani.mohamed-elhassaniSummary: Let \(G_2\) be a group which acts trivially on an abelian group \(G_1\). According to the Schreier's Theorem, each 2-cocycle \(\varepsilon \in Z^2(G_2,G_1)\) determines a group \(G_{\varepsilon}\) which is a central extension of \(G_1\) by \(G_2\), and we will denote this group by \(G_1\underset{\varepsilon}\times G_2\) and call it the perturbed direct product of \(G_1\) by \(G_2\) under \(\varepsilon\). The aim of this paper is to study properties of the perturbed direct products. For two distinct 2-cocycles \(\varepsilon_1\) and \(\varepsilon_2\), we find necessary and sufficient conditions for \(G_1\underset{\varepsilon_1}\times G_2\) to be isomorphic to \(G_1\underset{\varepsilon_2}\times G_2\). Furthermore, we obtain some results about decompositions for a given perturbed direct product \(G_1\underset{\varepsilon}{\times}G_2\) when \(G_1\) or \(G_2\) is a nontrivial direct product.Cellular covers of local groupshttps://zbmath.org/1491.201032022-09-13T20:28:31.338867Z"Flores, Ramón"https://zbmath.org/authors/?q=ai:flores.ramon-j"Scherer, Jérôme"https://zbmath.org/authors/?q=ai:scherer.jeromeSummary: We prove that, in the category of groups, the composition of a cellularization and a localization functor need not be idempotent. This provides a negative answer to a question of Emmanuel Dror Farjoun.The number of automorphism of a finite abelian group of rank twohttps://zbmath.org/1491.201042022-09-13T20:28:31.338867Z"Sehgal, Amit"https://zbmath.org/authors/?q=ai:sehgal.amit"Sehgal, Sarita"https://zbmath.org/authors/?q=ai:sehgal.sarita"Sharma, P. K."https://zbmath.org/authors/?q=ai:sharma.pradeep-kumar|sharma.powan-k|sharma.pawan-k|sharma.poonam-kumar|sharma.praveen-kumar|sharma.pramod-kumar(no abstract)Strongly indecomposable Butler groupshttps://zbmath.org/1491.201052022-09-13T20:28:31.338867Z"Blagoveshchenskaya, E."https://zbmath.org/authors/?q=ai:blagoveshchenskaya.ekaterina-anatolevnaSummary: We introduce a new class of torsion-free abelian groups which are special epimorphic images of finite rank completely decomposable groups. They belong to the well-known class of Butler groups by their definition. In comparison with the existing results the presented groups are not only of the maximal possible rank that is obviously less than the rank of the pre-image. The kernel of the above epimorphism admits a special matrix representation. The number of matrix rows is equal to the kernel rank and the number of its columns coincides with the preimage rank. By finitely many suitable operations the original matrix representation can be deduced to a special trapezoid form which corresponds to another choice of the epimorphism kernel generators. This matrix form clarifies the properties of the image that is exactly a group under investigation. For this class of torsion-free abelian groups of finite rank a strong indecomposability criterion is proved on the basis of their matrix representation.Completely \(0\)-simple semigroup with the basis propertyhttps://zbmath.org/1491.201062022-09-13T20:28:31.338867Z"Al-Dayel, Ibrahim"https://zbmath.org/authors/?q=ai:al-dayel.ibrahim"Al Khalaf, Ahmad"https://zbmath.org/authors/?q=ai:al-khalaf.ahmadOn dominions and varieties of bandshttps://zbmath.org/1491.201072022-09-13T20:28:31.338867Z"Abbas, Shabnam"https://zbmath.org/authors/?q=ai:abbas.shabnam"Ashraf, Wajih"https://zbmath.org/authors/?q=ai:ashraf.wajih"Shah, Aftab Hussain"https://zbmath.org/authors/?q=ai:shah.aftab-hussainJoin irreducible 2-testable semigroupshttps://zbmath.org/1491.201082022-09-13T20:28:31.338867Z"Lee, Edmond W. H."https://zbmath.org/authors/?q=ai:lee.edmond-w-hSummary: A nontrivial pseudovariety is join irreducible if whenever it is contained in the complete join of some collection of pseudovarieties, then it is contained in one of the pseudovarieties. A finite semigroup is join irreducible if it generates a join irreducible pseudovariety. The present article is concerned with semigroups that are 2-testable in the sense that they satisfy any equation formed by a pair of words that begin with the same variable, end with the same variable, and share the same set of factors of length two. The main objective is to show that there exist precisely seven join irreducible pseudovarieties of 2-testable semigroups. As a consequence, it is decidable in quadratic time if a finite 2-testable semigroup is join irreducible.Varieties of commutative semigroups closed under dominionshttps://zbmath.org/1491.201092022-09-13T20:28:31.338867Z"Shah, A. H."https://zbmath.org/authors/?q=ai:shah.aftab-hussain"Nabi, Muneer"https://zbmath.org/authors/?q=ai:nabi.muneer"Bano, Sakeena"https://zbmath.org/authors/?q=ai:bano.sakeenaOn semigroup constructions induced by commuting retractions on a sethttps://zbmath.org/1491.201102022-09-13T20:28:31.338867Z"Botur, Michal"https://zbmath.org/authors/?q=ai:botur.michalSummary: If \(\mathbf{G}=(G;\cdot)\) is a semigroup, \(I\) is arbitrary set and \(\lambda, \rho :I\rightarrow I\) are mappings satisfying the equalities \(\lambda \lambda =\lambda\), \(\rho\rho =\rho\) and \(\lambda\rho =\rho\lambda\) then we define the semigroup \((G^I,\times)\) where \((x\times y)(i) := x(\lambda i)\cdot y(\rho i)\). This construction gives rise to four covariant and two contravariant functors and constitute three adjoint situations. We apply this functors for finding representation theorems.The structure of certain F-abundant semigroupshttps://zbmath.org/1491.201112022-09-13T20:28:31.338867Z"Huang, Huawei"https://zbmath.org/authors/?q=ai:huang.huawei"Deng, Lunzhi"https://zbmath.org/authors/?q=ai:deng.lunzhi"Qu, Yunyun"https://zbmath.org/authors/?q=ai:qu.yunyun"Li, Chunhua"https://zbmath.org/authors/?q=ai:li.chunhua(no abstract)Semigroups of an inductive composition of termshttps://zbmath.org/1491.201122022-09-13T20:28:31.338867Z"Kitpratyakul, Pongsakorn"https://zbmath.org/authors/?q=ai:kitpratyakul.pongsakorn"Pibaljommee, Bundit"https://zbmath.org/authors/?q=ai:pibaljommee.bunditCongruences on \(\sim\) bisimple right type B \(\omega\)-semigroupshttps://zbmath.org/1491.201132022-09-13T20:28:31.338867Z"Li, Chunhua"https://zbmath.org/authors/?q=ai:li.chunhua"Xu, Baogen"https://zbmath.org/authors/?q=ai:xu.baogen"Huang, Huawei"https://zbmath.org/authors/?q=ai:huang.huawei(no abstract)Some properties of semigroups generated from a Cayley functionhttps://zbmath.org/1491.201142022-09-13T20:28:31.338867Z"Manavalan, Lejo J."https://zbmath.org/authors/?q=ai:manavalan.lejo-j"Romeo, P. G."https://zbmath.org/authors/?q=ai:romeo.parackal-govindanSummary: Any transformation on a set \(S\) is called a Cayley function on \(S\) if there exists a semigroup operation on \(S\) such that \(\beta\) is an inner-translation. In this paper we describe a method to generate a semigroup with \(k\) number of idempotents, study some properties of such semigroups like greens relations and bi-ordered sets.Isomorphism theorems for basic constructive algebraic structures with special emphasize on constructive semigroups with apartness -- an overviewhttps://zbmath.org/1491.201152022-09-13T20:28:31.338867Z"Mitrović, Melanija"https://zbmath.org/authors/?q=ai:mitrovic.melanija-s"Silvestrov, Sergei"https://zbmath.org/authors/?q=ai:silvestrov.sergei-dSummary: This overview is an introduction to the basic constructive algebraic structures with apartness with special emphasises on a set and semigroup with apartness. The main purpose of this paper, inspired by \textit{A. Bauer} [Bull. Am. Math. Soc., New Ser. 54, No. 3, 481--498 (2017; Zbl 1469.03171)], is to make some sort of understanding of constructive algebra in Bishop's style position for those (classical) algebraists as well as for the ones who apply algebraic knowledge who might wonder what is constructive algebra all about. Every effort has been made to produce a reasonably prepared text with such definite need. In the context of basic constructive algebraic structures constructive analogous of isomorphism theorems will be given. Following their development, two points of view on a given subject: classical and constructive will be considered. This overview is not, of course, a comprehensive one.
For the entire collection see [Zbl 1467.16001].More on the non-commutative epigroups which are not quasi-commutativehttps://zbmath.org/1491.201162022-09-13T20:28:31.338867Z"Soltani, P."https://zbmath.org/authors/?q=ai:soltani.p"Tehranian, A."https://zbmath.org/authors/?q=ai:tehranian.aboulfazl|tehranian.abolfazl|tehranian.aref|tehranian.abolfazi"Doostie, H."https://zbmath.org/authors/?q=ai:doostie.hossein"Sorouhesh, M. R."https://zbmath.org/authors/?q=ai:sorouhesh.mohammad-reza(no abstract)On endomorphisms of power-semigroupshttps://zbmath.org/1491.201172022-09-13T20:28:31.338867Z"Susanti, Yeni"https://zbmath.org/authors/?q=ai:susanti.yeni"Koppitz, Joerg"https://zbmath.org/authors/?q=ai:koppitz.jorgCubic quasi-ideal of semigroupshttps://zbmath.org/1491.201182022-09-13T20:28:31.338867Z"Gaketem, Thiti"https://zbmath.org/authors/?q=ai:gaketem.thitiThe concept of cubic quasi-ideal in semigroups is introduced in this paper and the author studies the basic properties of it. Moreover, he obtains some necessary and sufficient conditions for a cubic bi-ideal in a semigroup to be a cubic quasi-ideal. Several characterization theorems of regular and intra-regular semigroups in terms of cubic quasi-ideals are nicely presented in the paper. It would be desirable if some examples and counterexamples were included to justify and validate the results obtained.
Reviewer: Sanjib Kumar Datta (Kalyani)On almost interior ideals and weakly almost interior ideals of semigroupshttps://zbmath.org/1491.201192022-09-13T20:28:31.338867Z"Kaopusek, Nutsaree"https://zbmath.org/authors/?q=ai:kaopusek.nutsaree"Kaewnoi, Thananya"https://zbmath.org/authors/?q=ai:kaewnoi.thananya"Chinram, Ronnason"https://zbmath.org/authors/?q=ai:chinram.ronnasonSummary: The notion of almost ideals in semigroups was introduced by \textit{O. Grosek} and \textit{L. Satko} in 1980 [Semigroup Forum 20, 233--240 (1980; Zbl 0439.20045)]. In this paper, we define the notions of almost interior ideals and weakly almost interior ideals of semigroups by using the concepts of almost ideals and interior ideals of semigroups and investigate their properties.Comment on: ``Bi-interior ideals of semigroups''https://zbmath.org/1491.201202022-09-13T20:28:31.338867Z"Kehayopulu, Niovi"https://zbmath.org/authors/?q=ai:kehayopulu.nioviFrobenius R-variety of the numerical semigroups contained in a given onehttps://zbmath.org/1491.201212022-09-13T20:28:31.338867Z"Rosales, J. C."https://zbmath.org/authors/?q=ai:rosales.jose-carlos"Branco, M. B."https://zbmath.org/authors/?q=ai:branco.manuel-baptista|branco.manuel-batista"Traesel, M. A."https://zbmath.org/authors/?q=ai:traesel.marcio-andreSummary: Let \(\triangle\) be a numerical semigroup and \(\mathrm{R}(\triangle)=\{S \mid S \text{ is a numericalsemigroup and } S\subseteq \triangle\}\). We prove that \(\mathrm{R}(\triangle)\) is Frobenius R-variety that can be arranged in a tree rooted in \(\triangle\). We introduce the concepts of Frobenius and genus number of \(S\) restricted to \(\triangle\) (respectively \(\mathrm{F}_{\triangle}(S)\) and \(\mathrm{g}_{\triangle}(S)\)). We give formulas for \(\mathrm{F}_{\triangle}(S)\), \(\mathrm{g}_{\triangle}(S)\) and generalizations of the Amorós's and Wilf's conjecture. Moreover, we will show that most of the results of irreducibility can be generalized to \(\mathrm{R}(\triangle)\)-irreducibility.Endomorphisms of Clifford semigroups with injective structure homomorphismshttps://zbmath.org/1491.201222022-09-13T20:28:31.338867Z"Worawiset, S."https://zbmath.org/authors/?q=ai:worawiset.somnuek"Koppitz, J."https://zbmath.org/authors/?q=ai:koppitz.jorgSummary: In the present paper, we study semigroups of endomorphisms on Clifford semigroups with injective structure homomorphisms, where the semilattice has a least element. We describe such Clifford semigroups having a regular endomorphism monoid. If the endomorphism monoid on the Clifford semigroup is completely regular then the corresponding semilattice has at most two elements. We characterize all Clifford semigroups \(G_{\alpha}\cup G_{\beta}\) (\(\alpha >\beta\)) with an injective structure homomorphism, where \(G_{\alpha}\) has no proper subgroup, such that the endomorphism monoid is completely regular. In particular, we consider the case that the structure homomorphism is bijective.Regular elements and BQ-elements in generalized semigroups of \(\mathbb{Z}_n \)https://zbmath.org/1491.201232022-09-13T20:28:31.338867Z"Chinram, Ronnason"https://zbmath.org/authors/?q=ai:chinram.ronnason"Srivirod, Benya"https://zbmath.org/authors/?q=ai:srivirod.benya"Baupradist, Samruam"https://zbmath.org/authors/?q=ai:baupradist.samruam(no abstract)Homogeneity of inverse semigroupshttps://zbmath.org/1491.201242022-09-13T20:28:31.338867Z"Quinn-Gregson, Thomas"https://zbmath.org/authors/?q=ai:quinn-gregson.thomasThe algebraic structure of a semigroup of sets of transformations with restricted rangehttps://zbmath.org/1491.201252022-09-13T20:28:31.338867Z"Anantayasethi, Ananya"https://zbmath.org/authors/?q=ai:anantayasethi.ananya"Koppitz, Joerg"https://zbmath.org/authors/?q=ai:koppitz.jorgSummary: We study a semigroup which represents a semigroup of sets of Boolean functions on a finite set using the concept of transformations with restricted range. For this semigroup, we determine the algebraic structure. In particular, we characterize the (left, right, and two-sided) ideals and the Green's relations. Moreover, for each of the Green's relations, we provide the greatest included congruence.\(\mathscr{L}\)-, \(\mathscr{R}\)- and \(\mathscr{H}\)-cross-sections in the strong endomorphism semigroup of the undirected graphshttps://zbmath.org/1491.201262022-09-13T20:28:31.338867Z"Bondar, E. A."https://zbmath.org/authors/?q=ai:bondar.eugenija-aSummary: In the present paper we show that the strong endomorphism monoid of a finite undirected graph without multiply edges contains a unique \(\mathscr{R}\)-cross-section up to an isomorphism. We find necessary and sufficient conditions of an existence of \(\mathscr{H}\)-cross-sections and construct examples of \(\mathscr{L}\)-cross-sections. Also we prove that any \(\mathscr{L}\)-, \(\mathscr{R}\)- and \(\mathscr{H}\)-cross-section of the strong endomorphism semigroup is isomorphic to the direct product of the corresponding cross-sections in symmetric semigroups.Regularity of the semigroups of transformations with a fixed point sethttps://zbmath.org/1491.201272022-09-13T20:28:31.338867Z"Chinram, Ronnason"https://zbmath.org/authors/?q=ai:chinram.ronnason"Yonthanthum, Winita"https://zbmath.org/authors/?q=ai:yonthanthum.winitaSummary: For a nonempty set \(X\), let \(T(X)\) and \(P(X)\) denote respectively the full transformation semigroup on \(X\) and the partial transformation semigroup on \(X\). For a nonempty subset \(S\) of \(X\), let
\[
\begin{aligned}
& T_{\mathcal{F}(S)}(X) = \{\alpha \in T(X) \mid x\alpha = x \text{ for all } x \in S\},\\
& P_{\mathcal{F}(S)}(X) = \{\alpha \in P(X) \mid x\alpha = x \text{ for all } x \in \operatorname{dom} \alpha \cap S\}.
\end{aligned}
\]
Then \(T_{\mathcal{F}(S)}(X)\) is a regular subsemigroup of \(T(X)\) and \(P_{\mathcal{F}(S)}(X)\) is a subsemigroup of \(P(X)\) which need not be regular. In this paper, a necessary and sufficient condition for an element of \(P_{\mathcal{F}(S)}(X)\) to be regular is given. Furthermore, we characterize the left regular and right regular elements of the semigroups \(T_{\mathcal{F}(S)}(X)\) and \(P_{\mathcal{F}(S)}(X)\) and made use of these results to deduce the left regularity and right regularity of them.On relative ranks of the semigroup of orientation-preserving transformations on infinite chainshttps://zbmath.org/1491.201282022-09-13T20:28:31.338867Z"Dimitrova, Ilinka"https://zbmath.org/authors/?q=ai:dimitrova.ilinka"Koppitz, Jörg"https://zbmath.org/authors/?q=ai:koppitz.jorgGreen's relations in finite transformation semigroupshttps://zbmath.org/1491.201292022-09-13T20:28:31.338867Z"Fleischer, Lukas"https://zbmath.org/authors/?q=ai:fleischer.lukas"Kufleitner, Manfred"https://zbmath.org/authors/?q=ai:kufleitner.manfredSummary: We consider the complexity of Green's relations when the semigroup is given by transformations on a finite set. Green's relations can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes then correspond to the strongly connected components. It is not difficult to show that, in the worst case, the number of equivalence classes is in the same order of magnitude as the number of elements. Another important parameter is the maximal length of a chain of components. Our main contribution is an exponential lower bound for this parameter. There is a simple construction for an arbitrary set of generators. However, the proof for constant alphabet is rather involved. Our results also apply to automata and their syntactic semigroups.
For the entire collection see [Zbl 1362.68016].Corrigendum to: ``Second centralizers in the semigroup of injective transformations''https://zbmath.org/1491.201302022-09-13T20:28:31.338867Z"Konieczny, Janusz"https://zbmath.org/authors/?q=ai:konieczny.januszCorrigendum to the author's paper [ibid. 13, No. 4, Article ID 2050069, 11 p. (2020; Zbl 1443.20089)].On the large rank of certain semigroups of transformations on a finite chainhttps://zbmath.org/1491.201312022-09-13T20:28:31.338867Z"Kumar, Jitender"https://zbmath.org/authors/?q=ai:kumar.jitenderRegularity of a semigroup of transformations with restricted range that preserves an equivalence relation and a cross-sectionhttps://zbmath.org/1491.201322022-09-13T20:28:31.338867Z"Pookpienlert, Chollawat"https://zbmath.org/authors/?q=ai:pookpienlert.chollawat"Honyam, Preeyanuch"https://zbmath.org/authors/?q=ai:honyam.preeyanuch"Sanwong, Jintana"https://zbmath.org/authors/?q=ai:sanwong.jintanaSummary: For a fixed nonempty subset \(Y\) of \(X\), let \(T(X,Y)\) be the semigroup consisting of all transformations from \(X\) into \(Y\). Let \(\rho\) be an equivalence relation on \(X, \hat{\rho}\) the restriction of \(\rho\) on \(Y\) and \(R\) a cross-section of the partition \(Y/\hat{\rho} \). We define \[T(X,Y,\rho,R) = \{\alpha\in T(X,Y) : R\alpha\subseteq R~\text{and}~(a,b)\in \rho \Rightarrow (a\alpha,b\alpha)\in\rho\}.\] Then \(T(X,Y,\rho,R)\) is a subsemigroup of \(T(X,Y)\). In this paper, we describe regular elements in \(T(X,Y,\rho,R)\), characterize when \(T(X,Y,\rho,R)\) is a regular semigroup and investigate some classes of \(T(X,Y,\rho,R)\) such as completely regular and inverse from which the results on \(T(X,\rho,R)\) and \(T(X,Y)\) can be recaptured easily when taking \(Y=X\) and \(\rho\) to be the identity relation, respectively. Moreover, the description of unit-regularity on \(T(X,\rho,R)\) is obtained.Magnifying elements in linear transformation semigroups with restricted range which nullity or co-rank are infinitehttps://zbmath.org/1491.201332022-09-13T20:28:31.338867Z"Prakitsri, Pongsan"https://zbmath.org/authors/?q=ai:prakitsri.pongsanVarious regularities for semigroups of transformations on a finite set determined by a zig-zag orderhttps://zbmath.org/1491.201342022-09-13T20:28:31.338867Z"Prinyasart, Thanakorn"https://zbmath.org/authors/?q=ai:prinyasart.thanakorn"Srithus, Ratana"https://zbmath.org/authors/?q=ai:srithus.ratana"Nuntharatkul, Jirapa"https://zbmath.org/authors/?q=ai:nuntharatkul.jirapaThe \(\pi\)-semisimplicity of locally inverse semigroup algebrashttps://zbmath.org/1491.201352022-09-13T20:28:31.338867Z"Ji, Yingdan"https://zbmath.org/authors/?q=ai:ji.yingdanSummary: In this paper, we first characterize when a semigroup has completely 0-simple semigroup as its principal factors. Let \(R\) be a commutative ring with an identity, and let \(S\) be a locally inverse semigroup with the set of idempotents locally pseudofinite. Assume that the principal factors of \(S\) are all completely 0-simple. Then, we prove that the contracted semigroup algebra \(R_0[S]\) is \(\pi\)-semisimple if and only if the contracted semigroup algebras of all the principal factors of \(S\) are \(\pi\)-semisimple. Examples are provided to illustrate that the locally pseudofinite condition on the idempotent set of \(S\) cannot be removed. Notice that we extend the corresponding results on finite locally inverse semigroups.Connectivity, indecomposable, and weakly reversible in \(S\)-posetshttps://zbmath.org/1491.201362022-09-13T20:28:31.338867Z"Al Subaiei, Bana"https://zbmath.org/authors/?q=ai:al-subaiei.banaNoncommutative inclusion-exclusion, representations of left regular bands and the Tsetlin libraryhttps://zbmath.org/1491.201372022-09-13T20:28:31.338867Z"Blachar, Guy"https://zbmath.org/authors/?q=ai:blachar.guy"Rowen, Louis H."https://zbmath.org/authors/?q=ai:rowen.louis-halle"Vishne, Uzi"https://zbmath.org/authors/?q=ai:vishne.uziThe structure of regular right uniform semigroupshttps://zbmath.org/1491.201382022-09-13T20:28:31.338867Z"Hosseinzadeh Alikhalaji, Samira"https://zbmath.org/authors/?q=ai:hosseinzadeh-alikhalaji.samira"Sedaghatjoo, Mojtaba"https://zbmath.org/authors/?q=ai:sedaghatjoo.mojtaba"Roueentan, Mohammad"https://zbmath.org/authors/?q=ai:roueentan.mohammadSummary: In this paper, we investigate the right uniform notion on some classes of semigroups. The main objective of this paper is identifying the structure of regular right uniform semigroups which can be applied as a cornerstone of characterizing regular right subdirectly irreducible semigroups.Syntactic nondeterministic monoidshttps://zbmath.org/1491.201392022-09-13T20:28:31.338867Z"Kalampakas, Antonios"https://zbmath.org/authors/?q=ai:kalampakas.antonios"Louscou-Bozapalidou, Olympia"https://zbmath.org/authors/?q=ai:louscou-bozapalidou.olympia(no abstract)Semigroups of linear tree languageshttps://zbmath.org/1491.201402022-09-13T20:28:31.338867Z"Kitpratyakul, Pongsakorn"https://zbmath.org/authors/?q=ai:kitpratyakul.pongsakorn"Pibaljommee, Bundit"https://zbmath.org/authors/?q=ai:pibaljommee.bunditPrime ideals of gamma semigroups on weak nearness approximation spaceshttps://zbmath.org/1491.201412022-09-13T20:28:31.338867Z"Öztürk, Mehmet Ali"https://zbmath.org/authors/?q=ai:ozturk.mehmet-aliGamma ideal extension in gamma systemshttps://zbmath.org/1491.201422022-09-13T20:28:31.338867Z"Saied, Samir M."https://zbmath.org/authors/?q=ai:saied.samir-m"Khalil, Shuker Mahmood"https://zbmath.org/authors/?q=ai:khalil.shuker-mahmoodSummary: Let \(\gamma\) be a nonempty set, \(S^1\) be a commutative \(\gamma \)-semigroup contains \(\gamma \)-identity and \(M\) a unitary right \(S^1_\gamma\)-system (act). The notion of a \(\gamma \)-ideal extension on gamma systems are given and studied. An \(S^1_\gamma\)-system \(M\) is said to be a \(\gamma \)-ideal extension, if for each \(S^1_\gamma\)-homomorphism \(f: L \rightarrow M\) such that \(f(L)\subseteq M_\gamma L\), for each \(\gamma \)-ideal \(L\) of \(S^1\). A lot of characterizations and properties of gamma ideal extension of gamma systems have been given. In truth, we offer that \(\gamma \)-ideal extension of \(S^1_\gamma\)-system \(M\) is equal to the following similar terms 1. \(L_MR_{S^1}(\ell\gamma S^1))=M\gamma\ell,\forall\ell\in S^1\). 2. For all \(\ell\in S^1\) and \(m\in M\) \(R_{S^1}(\ell\gamma S^1)\subseteq R_{S^1}(m\gamma S^1)\) implies that \(m\in M\gamma\ell\). 3. For all \(\ell\in S^1\) and \(m\in M\), \([R_{S^1}(m\gamma S^1): R_{S^1}(\ell\gamma S^1)]=[M\gamma\ell:m\gamma S^1]\). 4. For all \(m\in M\), then \((L_M(\rho))=M\gamma\ell\), for some \(\gamma \)-compatible \(\rho\) on \(S^1\).On core quandles of groupshttps://zbmath.org/1491.201432022-09-13T20:28:31.338867Z"Bergman, George M."https://zbmath.org/authors/?q=ai:bergman.george-mSummary: We review the definition of a \textit{quandle}, and in particular of the \textit{core quandle} Core\((G)\) of a group \(G\), which consists of the underlying set of \(G\), with the binary operation \(x\lhd y=xy^{-1}x\) This is an \textit{involutory} quandle, i.e., satisfies the identity \(x\lhd(x\lhd y)=y\) in addition to the other identities defining a quandle.
\textit{Trajectories} \((x_i)_{i\in\mathbb{Z}}\) in groups and in involutory quandles (in the former context, sequences of the form \(x_ixz^i(x,z\in G)\) among other characterizations; in the latter, sequences satisfying \(x_{i+1}=x_i\lhd x_{i-1})\) are examined. A family of necessary conditions for an involutory quandle \(Q\) to be embeddable in the core quandle of a group is noted. Some implications are established between identities holding in groups and in their core quandles. Upper and lower bounds are obtained on the number of elements needed to generate the quandle Core\((G)\) for \(G\) a finitely generated group. Several questions are posed.On the classification of \(f\)-quandleshttps://zbmath.org/1491.201442022-09-13T20:28:31.338867Z"Churchill, Indu Rasika"https://zbmath.org/authors/?q=ai:churchill.indu-r-u"Elhamdadi, Mohamed"https://zbmath.org/authors/?q=ai:elhamdadi.mohamed"Van Kempen, Nicolas"https://zbmath.org/authors/?q=ai:van-kempen.nicolasSummary: We use the structural aspects of the \(f\)-quandle theory to classify, up to isomorphisms, all \(f\)-quandles of order \(n\). The classification is based on an effective algorithm that generate and check all \(f\)-quandles for a given order. We also include a pseudocode of the algorithm.
For the entire collection see [Zbl 1467.16001].A survey of racks and quandles: some recent developmentshttps://zbmath.org/1491.201452022-09-13T20:28:31.338867Z"Elhamdadi, Mohamed"https://zbmath.org/authors/?q=ai:elhamdadi.mohamedOn some associative formula with functional variableshttps://zbmath.org/1491.201462022-09-13T20:28:31.338867Z"Harutyunyan, D. N."https://zbmath.org/authors/?q=ai:harutyunyan.d-nSummary: In work of \textit{Yu. Movsisyan} and \textit{A. Gevorgyan} [Asian-Eur. J. Math. 14, No. 1, Article ID 2050155, 16 p. (2021; Zbl 1491.08002)], it was proved that the reversible algebra \((Q,\Sigma)\) satisfying the following second-order formula
\[\forall X,Y,\exists X',Y'\forall x,y,z(X(Y'(x,y),z)=Y(x,X'(y,z))),\]
is linear over the group. In the current work a more general result is proved that the regular and division algebra \((Q,\Sigma)\) with the mentioned second-order formula is endolinear over the group.Some applications of the independence to the semigroup of all binary systemshttps://zbmath.org/1491.201472022-09-13T20:28:31.338867Z"Rezaei, Akbar"https://zbmath.org/authors/?q=ai:rezaei.akbar"Kim, Hee Sik"https://zbmath.org/authors/?q=ai:kim.hee-sik"Neggers, Joseph"https://zbmath.org/authors/?q=ai:neggers.josephSummary: We extend the notions of right (left) independency and absorbent from groupoids to \(Bin(X)\) as a semigroup of all the groupoids on a set \(X\) and study and investigate many of their properties. We show that these new concepts are different by presenting several examples. In general, the concept of right (left) independence is a generalization and alternative of classical concept of the converse of injective function.Constructions of polynomially complete quasigroups of arbitrary orderhttps://zbmath.org/1491.201482022-09-13T20:28:31.338867Z"Artamonov, V. A."https://zbmath.org/authors/?q=ai:artamonov.vyacheslavovich-aleksandrovich|artamonov.vyacheslav-a"Chakrabarti, S."https://zbmath.org/authors/?q=ai:chakrabarti.subhadip|chakrabarti.sucharita|chakrabarti.sayak|chakrabarti.subhroneel|chakrabarti.soumen|chakrabarti.sukanya|chakrabarti.sayan-k|chakrabarti.satyabrata|chakrabarti.subrata-k|chakrabarti.sanjib-kumar|chakrabarti.subir-k|chakrabarti.sandip-k|chakrabarti.sadasiv|chakrabarti.saikat|chakrabarti.sulagna|chakrabarti.saswat|chakrabarti.sucheta|chakrabarti.soumya|chakrabarti.satish-chandra"Markov, V. T."https://zbmath.org/authors/?q=ai:markov.viktor-timofeevich"Pal, S. K."https://zbmath.org/authors/?q=ai:pal.swapan-kr|pal.surjya-k|pal.sayan-kumar|pal.surya-kant|pal.saibal-kumar|pal.sandip-kumar|pal.sudipta-kumar|pal.shiv-kumar|pal.sudip-kumar|pal.sankar-kr|pal.sankar-kumarOn subloops of the loop of invertible elements of the split Cayley-Dickson algebra over a field that contain a subloop of transvectionshttps://zbmath.org/1491.201492022-09-13T20:28:31.338867Z"Bashkirov, Evgenii L."https://zbmath.org/authors/?q=ai:bashkirov.evgenii-lIn [Commun. Algebra 49, No. 4, 1495--1512 (2021; Zbl 1479.20048)], the author studied the subset of invertible elements and the subset of elements of norm 1, both of which forming Moufang loops. This study, which has been started in [loc. cit.] is based on the fact that $G(K)$ contains the so-called transvections, i.e., elements $x\in O(K)$ satisfying the equation $[[(x-1)]]^2=0$. In this paper, the author continues the study of the subloop structure of the Moufang loop $G(K)$ formed by the invertible elements of the split Cayley-Dickson (octonion) algebra $O(K)$ over a field $K$. According to the author, the focus of this paper is on subloops of $G(K)$ that contain some, enough large, families of transvections, the total number of the families being, however small. The necessity for the restrictions indicated lies mainly in the fact that if $K$ is of characteristic not 2, then $G(K)$ contains a subloop isomorphic to the group of the unitriangular $3\times 3$ matrices over $K$. Furthermore, in this paper, the author investigates which members of the subloop lattice mentioned are distinct from each other in a loop-theoretical sense, that is, which of them are not isotopic.
Reviewer: C. Pereira da Silva (Curitiba)Mal'cev classes of left quasigroups and quandleshttps://zbmath.org/1491.201502022-09-13T20:28:31.338867Z"Bonatto, Marco"https://zbmath.org/authors/?q=ai:bonatto.marco"Fioravanti, Stefano"https://zbmath.org/authors/?q=ai:fioravanti.stefanoSummary: In this paper we investigate some Mal'cev classes of varieties of left quasigroups. We prove that the weakest non-trivial Mal'cev condition for a variety of left quasigroups is having a Mal'cev term and that every congruence meet-semidistributive variety of left quasigroups is congruence arithmetic. Then we specialize to the setting of quandles for which we prove that the congruence distributive varieties are those which have no non-trivial finite models.Every Moufang loop of odd order has nontrivial nucleushttps://zbmath.org/1491.201512022-09-13T20:28:31.338867Z"Csörgő, Piroska"https://zbmath.org/authors/?q=ai:csorgo.piroskaIn this paper, the author give a negative answer by proving that every Moufang loop of odd order has nontrivial nucleus. To obtain his result the author focus on Moufang loops satisfying any one of the three equivalent Moufang identities: $((xy)x)z = x(y(xz))$; $((xy)z)y = x(y(zy))$; $(xy)(zx) = x(yz)x$. A class of Moufang loop can be viewed as a ``group with weakened associativity''. The links between algebra, geometry, and group theory explain the importance of this class. Thus, according to the author, the main objective of the paper is to solve one of the main problems in the area of Moufang loops, namely: does there exist a Moufang loop of odd order with trivial nucleus? So, the author gives a negative answer. For the proof of the main result, the author used the structural properties of the multiplication group of Moufang loops of odd order. The author's proof is completely group theoretical relying on the theory of connected transversals. This concept was introduced by \textit{M. Niemenmaa} and \textit{T. Kepka} [J. Algebra 135, No. 1, 112--122 (1990; Zbl 0706.20046)]. Using their characterization theorem, the author transforms loop theoretical problems into group theoretical problems.
Reviewer: C. Pereira da Silva (Curitiba)Strong polynomial completeness of almost all quasigroupshttps://zbmath.org/1491.201522022-09-13T20:28:31.338867Z"Galatenko, A. V."https://zbmath.org/authors/?q=ai:galatenko.aleksei-vladimirovich"Galatenko, V. V."https://zbmath.org/authors/?q=ai:galatenko.vladimir-v"Pankrat'ev, A. E."https://zbmath.org/authors/?q=ai:pankratev.anton-evgenevichSummary: In the paper, it is proved that almost all quasigroups are strongly polynomially complete, i.e., are not isotopic to quasigroups that are not polynomially complete.Efficient verification of polynomial completeness of quasigroupshttps://zbmath.org/1491.201532022-09-13T20:28:31.338867Z"Galatenko, A. V."https://zbmath.org/authors/?q=ai:galatenko.aleksei-vladimirovich"Pankratiev, A. E."https://zbmath.org/authors/?q=ai:pankratev.anton-evgenevich"Staroverov, V. M."https://zbmath.org/authors/?q=ai:staroverov.vladimir-mikhailovichSummary: Polynomial completeness of an operation guarantees that deciding solvability of equations over this operation is an NP-complete problem. Thus this property is beneficial from the viewpoint of cryptographic applications. We propose an algorithm for verification of polynomial completeness of quasigroups and analyse efficiency of its serial and parallel implementations.Simple right conjugacy closed loopshttps://zbmath.org/1491.201542022-09-13T20:28:31.338867Z"Greer, Mark"https://zbmath.org/authors/?q=ai:greer.mark-rSummary: We give a general construction for right conjugacy closed loops, using \(\mathrm{GL}(2,q)\) for \(q\) a prime power. Under certain conditions, the loops constructed are simple, providing a general construction for finite, simple right conjugacy closed loops. Finally, a complete description of the isomorphism classes for the construction is given, yielding an exact count of non isomorphic loops for each \(q\).
For the entire collection see [Zbl 1410.17001].Smooth loops and loop bundleshttps://zbmath.org/1491.201552022-09-13T20:28:31.338867Z"Grigorian, Sergey"https://zbmath.org/authors/?q=ai:grigorian.sergeySummary: A loop is a rather general algebraic structure that has an identity element and division, but is not necessarily associative. Smooth loops are a direct generalization of Lie groups. A key example of a non-Lie smooth loop is the loop of unit octonions. In this paper, we study properties of smooth loops and their associated tangent algebras, including a loop analog of the Maurer-Cartan equation. Then, given a manifold, we introduce a loop bundle as an associated bundle to a particular principal bundle. Given a connection on the principal bundle, we define the torsion of a loop bundle structure and show how it relates to the curvature, and also develop aspects of a non-associative gauge theory. Throughout, we see how some of the known properties of \(G_2\)-structures can be seen from this more general setting.Latin quandles and applications to cryptographyhttps://zbmath.org/1491.201562022-09-13T20:28:31.338867Z"Isere, Abednego Orobosa"https://zbmath.org/authors/?q=ai:isere.abednego-orobosa"Adéníran, John Olúsọlá"https://zbmath.org/authors/?q=ai:adeniran.john-olusola"Jaíyéọlá, Tèmít\'ọp\'ẹ Gb\'ọláhàn"https://zbmath.org/authors/?q=ai:jaiyeola.temitope-gbolahanSummary: This work investigated some properties of Latin quandles that are applicable in cryptography. Four distinct cores of an Osborn loop (non-diassociative and non-power associative) were introduced and investigated. The necessary and sufficient conditions for these cores to be (i) (left) quandles (ii) involutory quandles (iii) quasi-Latin quandles and (iv) involutory quasi-Latin quandles were established. These conditions were judiciously used to build cipher algorithms for cryptography in some peculiar circumstances.New algebraic properties of middle Bol loops. IIhttps://zbmath.org/1491.201572022-09-13T20:28:31.338867Z"Jaiyéolá, T. G."https://zbmath.org/authors/?q=ai:jaiyeola.temitope-gbolahan"David, S. P."https://zbmath.org/authors/?q=ai:david.sunday-peter"Oyebola, O. O."https://zbmath.org/authors/?q=ai:oyebola.oyeyemi-oSummary: A loop \((Q, \cdot, \backslash, /)\) is called a middle Bol loop (MBL) if it obeys the identity \(x(yz \backslash x)=(x/z)(y \backslash x)\). To every MBL corresponds a right Bol loop (RBL) and a left Bol loop (LBL). In this paper, some new algebraic properties of a middle Bol loop are established in a different style. Some new methods of constructing a MBL by using a non-abelian group, the holomorph of a right Bol loop and a ring are described. Some equivalent necessary and sufficient conditions for a right (left) Bol loop to be a middle Bol loop are established. A RBL (MBL, LBL, MBL) is shown to be a MBL (RBL, MBL, LBL) if and only if it is a Moufang loop.
For Part I see [the first author et al., ROMAI J. 11, No. 2, 161--183 (2015; Zbl 1424.20088)].Basarab loop and the generators of its total multiplication grouphttps://zbmath.org/1491.201582022-09-13T20:28:31.338867Z"Jaiyéọlá, T. G."https://zbmath.org/authors/?q=ai:jaiyeola.temitope-gbolahan"Effiong, G. O."https://zbmath.org/authors/?q=ai:effiong.gideon-okonSummary: A loop \((Q, \cdot )\) is called a Basarab loop if the identities: \((x \cdot yx^\rho )(xz) = x \cdot yz\) and \((yx)\cdot (x^\lambda z \cdot x) = yz \cdot x\) hold. It was shown that the left, right and middle nuclei of the Basarab loop coincide, and the nucleus of a Basarab loop is the set of elements \(x\) whose middle inner mapping \(T_x\) are automorphisms. The generators of the inner mapping group of a Basarab loop were refined in terms of one of the generators of the total inner mapping group of a Basarab loop. Necessary and sufficient condition(s) in terms of the inner mapping group (associators) for a loop to be a Basarab loop were established. It was discovered that in a Basarab loop: the mapping \(x \mapsto T_x\) is an endomorphism if and only if the left (right) inner mapping is a left (right) regular mapping. It was established that a Basarab loop is a left and right automorphic loop and that the left and right inner mappings belong to its middle inner mapping group. A Basarab loop was shown to be an automorphic loop (A-loop) if and only if it is a middle automorphic loop (middle Aloop). Some interesting relations involving the generators of the total multiplication group and total inner mapping group of a Basarab loop were derived, and based on these, the generators of the total inner mapping group of a Basarab loop were finetuned. A Basarab loop was shown to be a totally automorphic loop (TA-loop) if and only if it is a commutative and flexible loop. These aforementioned results were used to give a partial answer to a 2013 question and an ostensible solution to a 2015 problem in the case of Basarab loop.Classifying gyrotransversals in groupshttps://zbmath.org/1491.201592022-09-13T20:28:31.338867Z"Lal, Ratan"https://zbmath.org/authors/?q=ai:lal.ratan"Kakkar, Vipul"https://zbmath.org/authors/?q=ai:kakkar.vipulNon-commuting graphs and some bounds for commutativity degree of finite Moufang loopshttps://zbmath.org/1491.201602022-09-13T20:28:31.338867Z"Rezaie, Elhameh"https://zbmath.org/authors/?q=ai:rezaie.elhameh"Ahmadidelir, Karim"https://zbmath.org/authors/?q=ai:ahmadidelir.karim"Tehranian, Abolfazl"https://zbmath.org/authors/?q=ai:tehranian.abolfazl"Rasouli, Hamid"https://zbmath.org/authors/?q=ai:rasouli.hamidSummary: In this paper, we study some properties of the non-commuting graph \(\varGamma_M\) of a finite Moufang loop \(M\), a graph obtained by setting all non-central elements of \(M\) as the vertex set and defining two distinct vertices to be adjacent if and only if their commutator is non-identity. In particular, Hamiltonian as well as (weak) perfectness of non-commuting graphs of Chein loops are considered. We find several upper and lower bounds for commutativity degree of some classes of finite Moufang loops by means of edge number of their non-commuting graphs and algebraic properties.Semi-lattice of varieties of quasigroups with linearityhttps://zbmath.org/1491.201612022-09-13T20:28:31.338867Z"Sokhatsky, F. M."https://zbmath.org/authors/?q=ai:sokhatsky.fedir-m"Krainichuk, H. V."https://zbmath.org/authors/?q=ai:krainichuk.halyna-v"Sydoruk, V. A."https://zbmath.org/authors/?q=ai:sydoruk.v-aSummary: A \(\sigma\)-parastrophe of a class of quasigroups \(\mathfrak{A}\) is a class \({^{\sigma}\mathfrak{A}}\) of all \(\sigma\)-parastrophes of quasigroups from \(\mathfrak{A}\). A set of all pairwise parastrophic classes is called a \textit{parastrophic orbit} or a \textit{truss}. A parastrophically closed semi-lattice of classes is a \textit{bunch}.
A \textit{linearity bunch} is a set of varieties which contains the variety of all left linear quasigroups, the variety of all left alinear quasigroups, all their parastrophes and all their intersections. It contains 14 varieties, which are distributed into six parastrophic orbits. All quasigroups from these varieties are called dilinear. To obtain all varieties from the bunch, concepts of \textit{middle linearity} and \textit{middle alinearity} are introduced. A well-known identity or a system of identities which describes a variety from every parastrophic orbit of the bunch is cited. An algorithm for obtaining identities which describe all varieties from the parastrophic orbits is given. Examples of quasigroups distinguishing one variety from the other are presented.Geometry of gyrogroups via Klein's approachhttps://zbmath.org/1491.201622022-09-13T20:28:31.338867Z"Suksumran, Teerapong"https://zbmath.org/authors/?q=ai:suksumran.teerapongSummary: Using Klein's approach, geometry can be studied in terms of a space of points and a group of transformations of that space. This allows us to apply algebraic tools in studying geometry of mathematical structures. In this article, we follow Klein's approach to study the geometry \((G, \mathcal{T})\), where \(G\) is an abstract gyrogroup and \(\mathcal{T}\) is an appropriate group of transformations containing all gyroautomorphisms of \(G\). We focus on \(n\)-transitivity of gyrogroups and also give a few characterizations of coset spaces to be minimally invariant sets. We then prove that the collection of open balls of equal radius is a minimally invariant set of the geometry \((G, \Gamma_m)\) for any normed gyrogroup \(G\), where \(\Gamma_m\) is a suitable group of isometries of \(G\).Extension of the Švarc-Milnor lemma to gyrogroupshttps://zbmath.org/1491.201632022-09-13T20:28:31.338867Z"Wattanapan, Jaturon"https://zbmath.org/authors/?q=ai:wattanapan.jaturon"Atiponrat, Watchareepan"https://zbmath.org/authors/?q=ai:atiponrat.watchareepan"Suksumran, Teerapong"https://zbmath.org/authors/?q=ai:suksumran.teerapongSummary: A strongly generated gyrogroup is a gyrogroup that comes with a specific generating set invariant under its gyroautomorphisms, which may be viewed as a suitable generalization of a group. The achievement of constructing a word metric on a strongly generated gyrogroup, together with the recent notion of a gyrogroup action, prompts us to extend the Švarc-Milnor lemma (also known as the fundamental lemma of geometric group theory) to the case of gyrogroups.Minimality of pair \((I,J)\)-ideals in ternary semigroupshttps://zbmath.org/1491.201642022-09-13T20:28:31.338867Z"Dehghanian, Mehdi"https://zbmath.org/authors/?q=ai:dehghanian.mehdi"Mohammadhasani, Ahmad"https://zbmath.org/authors/?q=ai:mohammadhasani.ahmadSummary: Let \(S\) be a ternary semigroup. In this article we introduce our notation and prove some elementary properties of a pair ideal \((I,J)\) of a ternary semigroup \(S\) and give some characterizations of the minimality of pair left (right) and middle ideal in ternary semigroup.Reducibility of \(n\)-ary semigroups: from quasitriviality towards idempotencyhttps://zbmath.org/1491.201652022-09-13T20:28:31.338867Z"Couceiro, Miguel"https://zbmath.org/authors/?q=ai:couceiro.miguel"Devillet, Jimmy"https://zbmath.org/authors/?q=ai:devillet.jimmy"Marichal, Jean-Luc"https://zbmath.org/authors/?q=ai:marichal.jean-luc"Mathonet, Pierre"https://zbmath.org/authors/?q=ai:mathonet.pierreDenote \({x_1},\dots, {x_n}\) as \(x_1^n\). An \(n\)-ary (\(n \geqslant 2\)) operation \(F:{X^n} \to X\) on an nonempty set \(X\) is associative iff \(F(x_1^{i - 1},F(x_i^{i + n - 1}),x_{i + n}^{2n - 1}) = F(x_1^i,F(x_{i + 1}^{i + n}),x_{i + n + 1}^{2n - 1})\) for all \(x_1^{2n - 1} \in X\), \(1 \leqslant i \leqslant n - 1\); \((X,F)\) is then called \(n\)-ary semigroup. The operation \(F\) is reducible to binary operation \(G:{X^2} \to X\) if there are \({G^m}:{X^{m + 1}} \to X\), \(m = 1,\dots ,n - 1\) such that \({G^{n - 1}} = F\), \({G^1} = G\) and \({G^m}(x_1^{m + 1}) = {G^{m - 1}}(x_1^{m - 1}G(x_m^{m + 1}))\) for any \(m \geqslant 2\). Let \({\mathcal F}_k^n\) be the class of associative operations with property: \(F(x_1^n) \in \{ {x_1},\dots ,{x_n}\} \) if at least \(k\) of \({x_i}\)-s are equal. It is shown that \({\mathcal F}_1^n =\dots = {\mathcal F}_{n - 2}^n \subseteq {\mathcal F}_{n - 1}^n \subseteq {\mathcal F}_n^n\) and found conditions for the last two inclusions to be strict.
Reviewer: Jaak Henno (Tallinn)Partial Menger algebras of termshttps://zbmath.org/1491.201662022-09-13T20:28:31.338867Z"Denecke, K."https://zbmath.org/authors/?q=ai:denecke.klaus"Hounnon, H."https://zbmath.org/authors/?q=ai:hounnon.hippolyteOn the structure of symmetric \(n\)-ary bandshttps://zbmath.org/1491.201672022-09-13T20:28:31.338867Z"Devillet, Jimmy"https://zbmath.org/authors/?q=ai:devillet.jimmy"Mathonet, Pierre"https://zbmath.org/authors/?q=ai:mathonet.pierreMenger algebras of \(k\)-commutative \(n\)-place functionshttps://zbmath.org/1491.201682022-09-13T20:28:31.338867Z"Dudek, Wieslaw A."https://zbmath.org/authors/?q=ai:dudek.wieslaw-aleksander"Trokhimenko, Valentin S."https://zbmath.org/authors/?q=ai:trokhimenko.valentin-sLet \({g_1},\dots ,{g_n} = g_1^n\), \(g,\dots ,g = \mathop g\limits^n \). If \({\mathcal T}({A^n},A)\) is the set of all \(n\)-place everywhere defined functions on a nonempty set \(A\) and the \(n + 1\)-ary Menger superposition operation is defined by \({\mathcal O}(f,g_{_1}^n)(x_1^n) = f(g_1^n(x_1^n))\) then every subset \(\Phi \subseteq {\mathcal T}({A^n},A)\) closed for \({\mathcal O}\) is a Menger algebra \((\Phi ,{\mathcal O})\) of rank \(n\); alternative notation for Menger superposition is \({\mathcal O}(f,g_1^n) = f[g_1^n]\). An \(n\)-place function \(f \in {\mathcal T}({A^n},A)\) is \(k\)-commutative, \(1 \leqslant k \leqslant n - 1\), if for all \(a,b \in A\) and all \(i = 1,2,\dots ,n - k\) holds\(f(\mathop a\limits^{i - 1} ,\mathop b\limits^k ,\mathop a\limits^{n - i - k + 1} ) = f(\mathop b\limits^{i - 1} ,\mathop a\limits^k ,\mathop b\limits^{n - i - k + 1} )\); \(\mathfrak{M}_n^k(A)\) is the set of all \(k\)-commutative \(n\)-place functions on the set \(A\), \((\mathfrak{M}_n^k(A),{\mathcal O})\) is a subalgebra of \(({\mathcal T}({A^n},A),{\mathcal O})\). It is shown that a Menger algebra \((G,o)\) of rank \(n \geqslant 2\) can be isomorphically embedded into a Menger algebra \((\mathfrak{M}_n^k(A),{\mathcal O})\) iff \(x[\mathop y\limits^{i - 1} \mathop z\limits^k \mathop y\limits^{n - i - k + 1} ] = x[\mathop z\limits^{i - 1} \mathop y\limits^k \mathop z\limits^{n - i - k + 1} ]\).
A subset \(H\) of a Menger algebra \((G,o)\) is
\begin{itemize}
\item[(1)] an \(s\)-ideal iff \((\forall h,x_1^n \in G)(h \in H \Rightarrow h[x_1^n] \in H)\);
\item[(2)] a \(v\)-ideal iff \((\forall x,h_1^n \in G)(h_1^n \in H \Rightarrow x[h_1^n] \in H)\);
\item[(3)] an \(sv\)-ideal iff it is both an \(s\)- and a \(v\)-ideal.
\end{itemize}
Denote by \({C_A} = \{ {f_a}:a \in A,(\forall {a_1},\dots ,{a_n} \in A)f(a_1^n) = a\} \) the set of all constant functions. It is shown that \({C_A}\) is densely embedded in \((\mathfrak{M}_n^k(A),{\mathcal O})\) (every nontrivial congruence on parent induces a nontrivial congruence on \({C_A}\)). The set of constants and dense embeddings are used to present a description of all Menger algebras isomorphic to some \((\mathfrak{M}_n^k(A),{\mathcal O})\) and to prove, that all automorphisms of \((\mathfrak{M}_n^k(A),{\mathcal O})\) are inner.
Reviewer: Jaak Henno (Tallinn)Left translations and isomorphism theorems for Menger algebras of rank \(n\)https://zbmath.org/1491.201692022-09-13T20:28:31.338867Z"Kumduang, Thodsaporn"https://zbmath.org/authors/?q=ai:kumduang.thodsaporn"Leeratanavalee, Sorasak"https://zbmath.org/authors/?q=ai:leeratanavalee.sorasakSummary: Let \(n\) be a fixed natural number. Menger algebras of rank \(n\) can be regarded as a canonical generalization of arbitrary semigroups. This paper is concerned with studying algebraic properties of Menger algebras of rank \(n\) by first defining a special class of full \(n\)-place functions, the so-called a left translation, which possess necessary and sufficient conditions for an \((n+1)\)-groupoid to be a Menger algebra of rank \(n\). The isomorphism parts begin with introducing the concept of homomorphisms, and congruences in Menger algebras of rank \(n\). These lead us to establish a quotient structure consisting a nonempty set factored by such congruences together with an operation defined on its equivalence classes. Finally, the fundamental homomorphism theorem and isomorphism theorems for Menger algebras of rank \(n\) are given. As a consequence, our results are significant in the study of algebraic theoretical Menger algebras of rank \(n\). Furthermore, we extend the usual notions of ordinary semigroups in a natural way.Left regular and left weakly regular \(n\)-ary semigroupshttps://zbmath.org/1491.201702022-09-13T20:28:31.338867Z"Pornsurat, Patchara"https://zbmath.org/authors/?q=ai:pornsurat.patchara"Pibaljommee, Bundit"https://zbmath.org/authors/?q=ai:pibaljommee.bunditThe concept of quasi-ideals of an \(n\)-ary semigroupis introduced and the new concept of regularity (different from classical [\textit{F. M. Sioson}, Proc. Japan Acad. 39, 283--286 (1963; Zbl 0118.02104)] is introduced. Now an \(n\)-ary semigroup \((G,f)\) is left regular if for every \(a\in S\) there are \(x_i\in S\) such that \(a=f(x_1,\ldots,x_{n-1},f(a,a,\ldots,a))\). Such defined regularity is characterized by quasi-ideals.
Reviewer: Wiesław A. Dudek (Wrocław)A hyperstructures approach to the direct limit and tensor product for left(right) \(G\)-sets on hypergroupshttps://zbmath.org/1491.201712022-09-13T20:28:31.338867Z"Khorshid, Noreh Rakhsh"https://zbmath.org/authors/?q=ai:khorshid.noreh-rakhsh"Ostadhadi-Dehkordi, Sohrab"https://zbmath.org/authors/?q=ai:dehkordi.sohrab-ostadhadiSummary: In this paper, we introduce the concept left(right)-\(G\) sets by external hyperoperations and some examples presented. We construct quotient left(right)-\(G\) sets by regular (strongly) regular relations. Also, we consider fundamental relation as a smallest strongly regular relations and by complete parts concepts introduce an equivalence relation that is coincide with fundamental relation. The main purpose of this paper is to introduce the concepts of tensor product and direct limit on \(G\)-sets of \(n\)-ary semihypergroups that are non-additive modication of classical construction in module theory. This concept is crucially important in homological algebra and several properties are found and examples are presented.Hypergroups associated with hypergraphshttps://zbmath.org/1491.201722022-09-13T20:28:31.338867Z"Nikkhah, A."https://zbmath.org/authors/?q=ai:nikkhah.abolfazl"Davvaz, B."https://zbmath.org/authors/?q=ai:davvaz.bijanQuasi-order \(\Gamma\)-semihypergroups: fundamental relations and complete partshttps://zbmath.org/1491.201732022-09-13T20:28:31.338867Z"Ostadhadi-Dehkordi, S."https://zbmath.org/authors/?q=ai:ostadhadi-dehkordi.s"Shum, K. P."https://zbmath.org/authors/?q=ai:shum.kar-pingOn almost quasi-hyperideals in semihypergroupshttps://zbmath.org/1491.201742022-09-13T20:28:31.338867Z"Suebsung, Sudaporn"https://zbmath.org/authors/?q=ai:suebsung.sudaporn"Yonthanthum, Winita"https://zbmath.org/authors/?q=ai:yonthanthum.winita"Hila, Kostaq"https://zbmath.org/authors/?q=ai:hila.kostaq"Chinram, Ronnason"https://zbmath.org/authors/?q=ai:chinram.ronnasonSummary: In this paper, we define almost quasi-hyperideals in semihypergroups and give some interesting properties.Solvable hypergroups and a generalization of Hall's theorems on finite solvable groups to association schemeshttps://zbmath.org/1491.201752022-09-13T20:28:31.338867Z"Vasil'ev, Andrey V."https://zbmath.org/authors/?q=ai:vasilev.andrei-viktorovich"Zieschang, Paul-Hermann"https://zbmath.org/authors/?q=ai:zieschang.paul-hermannSummary: Within the class of hypergroups (in the sense of \textit{F. Marty} [in: 8. Skand. Mat. Kongr., Stockholm, 45--49 (1934; JFM 61.1014.03)]) we consider a specific class of hypergroups the members of which we call solvable. Our definition of solvable hypergroups generalizes the notion of a solvable group. We investigate solvable hypergroups and utilize our results to generalize Philip Hall's celebrated theorems on finite solvable groups to the theory of association schemes.Multi-fuzzy soft groupshttps://zbmath.org/1491.201762022-09-13T20:28:31.338867Z"Akin, Canan"https://zbmath.org/authors/?q=ai:akin.cananSummary: The purpose of this study is to extend the multi-fuzzy soft sets to the theory of groups. The concept of multi-fuzzy soft sets is examined and some new definitions are introduced such as restricted and extended intersection, restricted and extended union, \( \wedge \)-intersection and \(\vee \)-union of the family multi-fuzzy soft sets and also a definition of a multi-fuzzy soft function is given. In this paper, the concept of multi-fuzzy soft sets is applied to the theory of groups and a notion which is called a multi-fuzzy soft group is proposed. A definition of homomorphism of multi-fuzzy soft groups is presented. We obtain that the image (or pre-image) of a multi-fuzzy soft group is a multi-fuzzy soft group.On fundamental isomorphism theorems in soft subgroupshttps://zbmath.org/1491.201772022-09-13T20:28:31.338867Z"Çağman, N."https://zbmath.org/authors/?q=ai:cagman.naim"Barzegar, R."https://zbmath.org/authors/?q=ai:barzegar.r"Hosseini, S. B."https://zbmath.org/authors/?q=ai:hosseini.s-bSummary: Molodsov initiated a novel concept of soft set theory, which is a completely new approach for modeling vagueness and uncertainty, which there is no limited condition to description of objects and is free from the difficulties affecting existing methods. This makes the theory very convenient and easy to apply in practice. After the pioneering work of Molodsov, there has been a great effort to obtain soft set analogues of classical theories. Among other fields, a progressive developments are made in the field of algebraic structure. To extend the soft set in group theory, many researchers introduced the notions of soft subgroup and investigated its applications in group theory and decision making. In this paper, by using the soft sets and their duality, we introduce new concepts on the soft sets, which are called soft quotient subgroup and quotient dual soft subgroup. We then derive their algebraic properties and, in sequel, investigate the fundamental isomorphism theorems in soft subgroups analogous to the group theory.On interval valued fuzzy almost \((m, n)\)-ideal in semigroupshttps://zbmath.org/1491.201782022-09-13T20:28:31.338867Z"Gaketem, Thiti"https://zbmath.org/authors/?q=ai:gaketem.thiti(no abstract)Extensions of soft ideals over semigroupshttps://zbmath.org/1491.201792022-09-13T20:28:31.338867Z"Hamouda, Essam"https://zbmath.org/authors/?q=ai:hamouda.essam-h"Ramadan, A. A."https://zbmath.org/authors/?q=ai:ramadan.ahmed-abd-el-kader"Amira, S."https://zbmath.org/authors/?q=ai:amira.sSummary: Let \((F,A)\) be a soft ideal over a semigroup \(S\). We introduce the concept of the extension of \((F,A)\). By using the mapping \(\alpha_F\) corresponding to \((F,A)\), we define the notions \(\alpha\)-prime and \(\alpha\)- semiprime soft sets over \(S\). Relationships between these concepts are investigated and illustrative examples are provided. Also, by means of the extensions of a soft ideal \((F,A)\), a congruence relation \(\eta_F\) on \(\mathrm{S}\) is defined. It is shown that \(\eta_F\) is a semilattice congruence if \((F,A)\) is a \(\alpha\)-semiprime soft left ideal over \(S\). Finally, the soft sets \((i_F,A)\) and \((I_F,A)\) over \(S\) are defined in terms of the concept of soft points.Counting distinct fuzzy subgroups of finite abelian groups of order \(p^nq^m\)https://zbmath.org/1491.201802022-09-13T20:28:31.338867Z"Han, Lingling"https://zbmath.org/authors/?q=ai:han.lingling"Guo, Xiuyun"https://zbmath.org/authors/?q=ai:guo.xiuyunSummary: The purpose of this paper is to count the distinct fuzzy subgroups of a finite abelian group of order \(p^nq^m\) for any different primes \(p,q\) and any positive integers \(n,m\). This counting problem is reduced to finite anelian \(p\)-groups. As applications of our main result, explicit formulas for the number of distinct fuzzy subgroups of the following two classes of finite abelian groups are given:
\begin{itemize}
\item[i)] The direct product \(\mathbb{Z}^n_p\times\mathbb{Z}^m_q\) of a finite elementary abelian \(p\)-group \(\mathbb{Z}^n_p\) and a finite elementary abelian \(q\)-group \(\mathbb{Z}^m_q\) with different primes \(p\) and \(q\);
\item[ii)] The direct product \(\mathbb{Z}^n_p\times\mathbb{Z}^m_q\) of a finite cyclic \(p\)-group \(\mathbb{Z}^p_n\) and a finite elementary abelian \(q\)-group \(\mathbb{Z}^m_q\) with different primes \(p\) and \(q\).
\end{itemize}Atanassov's intuitionistic anti fuzzy interior ideals of semigroupshttps://zbmath.org/1491.201812022-09-13T20:28:31.338867Z"Majumder, Samit Kumar"https://zbmath.org/authors/?q=ai:majumder.samit-kumarSummary: In this paper the concepts of intuitionistic anti fuzzy interior ideal in semigroups is introduced and some important characterizations have been obtained.Group operations and isomorphic relation with the 2-tuple linguistic variableshttps://zbmath.org/1491.201822022-09-13T20:28:31.338867Z"Malhotra, Tanya"https://zbmath.org/authors/?q=ai:malhotra.tanya"Gupta, Anjana"https://zbmath.org/authors/?q=ai:gupta.anjanaSummary: This paper aims to put forth the theory of 2-tuple linguistic groups concerning the binary operation in the conventional sense. For this, a formal methodology has been introduced to prove that a predefined nonempty linguistic term set, \textit{LT}, and the interval, \([\frac{-1}{2},\frac{1}{2}]\), forms a group. Further, we have proved that a set of all 2-tuple linguistic information, \( \overline{LT} \equiv LT \times [\frac{-1}{2},\frac{1}{2}]\), and numerical interval, \([-n,n]\), where \(n\) is presumed to be a positive integer, also forms a group. Later on, we develop a one-to-one correspondence and homomorphic group relation between the set of all 2-tuple linguistic information and numerical interval, \([-n,n]\). Henceforth, a similarity relation between the two groups is obtained. Finally, a practical application is defined by proposing the notion of a 2-tuple linguistic bipolar graph to illustrate the usefulness and practicality of the group isomorphic relation.On fundamental algebraic characterizations of \(\mu\)-fuzzy normal subgroupshttps://zbmath.org/1491.201832022-09-13T20:28:31.338867Z"Masmali, Ibtisam"https://zbmath.org/authors/?q=ai:masmali.ibtisam-a-h"Shuaib, Umer"https://zbmath.org/authors/?q=ai:shuaib.umer"Razaq, Abdul"https://zbmath.org/authors/?q=ai:razaq.abdul"Fatima, Areeba"https://zbmath.org/authors/?q=ai:fatima.areeba"Alhamzi, Ghaliah"https://zbmath.org/authors/?q=ai:alhamzi.ghaliah(no abstract)Some fuzzy multigroups obtained from fuzzy subgroupshttps://zbmath.org/1491.201842022-09-13T20:28:31.338867Z"Mirvakili, Saeed"https://zbmath.org/authors/?q=ai:mirvakili.saeed"Naraghi, Hossein"https://zbmath.org/authors/?q=ai:naraghi.hossein"Shirvani, Maryam"https://zbmath.org/authors/?q=ai:shirvani.maryam"Ghiasvand, Peyman"https://zbmath.org/authors/?q=ai:ghiasvand.peymanSummary: In this paper, first we produce a fuzzy multigroup from a fuzzy subgroup of a group \(G\) and we show that there is a relationship between a fuzzy multigroup with underlying group \(G\) and a fuzzy multigroup with underlying group \(Aut(A)\). Moreover, we generate a code by using the defined special fuzzy multigroup automorphisms.Categorical structures of soft groupshttps://zbmath.org/1491.201852022-09-13T20:28:31.338867Z"Öztunç, Simge"https://zbmath.org/authors/?q=ai:oztunc.simge"Aslan, Sedat"https://zbmath.org/authors/?q=ai:aslan.sedat"Dutta, Hemen"https://zbmath.org/authors/?q=ai:dutta.hemenSummary: In the current paper, the category of soft groups and soft group homomorphisms is constructed and it is proved that this structure satisfies the category conditions. Also, algebraic properties of some types of soft group morphisms are obtained. Finally, an application is presented as `\textit{cube}' of soft groups and soft group homomorphisms.Partial associativity and rough approximate groupshttps://zbmath.org/1491.201862022-09-13T20:28:31.338867Z"Gowers, W. T."https://zbmath.org/authors/?q=ai:gowers.william-timothy"Long, J."https://zbmath.org/authors/?q=ai:long.jianyong|long.jie|long.jianwu|long.jeff|long.jianceng|long.jinling|long.junyun|long.jingfan|long.jianmin|long.jiang|long.jianzhong|long.jason|long.jinghua|long.jiangqi|long.jiancheng|long.jun|long.junsheng|long.junbo|long.jiaping|long.jancis|long.jianyu|long.junyan|long.jing|long.jianhui|long.jane|long.jianjun|long.jihaoSummary: Suppose that a binary operation \(\circ\) on a finite set \(X\) is injective in each variable separately and also associative. It is easy to prove that \((X, \circ)\) must be a group. In this paper we examine what happens if one knows only that a positive proportion of the triples \((x, y, z) \in X^3\) satisfy the equation \(x \circ (y \circ z) = (x \circ y) \circ z\). Other results in additive combinatorics would lead one to expect that there must be an underlying `group-like' structure that is responsible for the large number of associative triples. We prove that this is indeed the case: there must be a proportional-sized subset of the multiplication table that approximately agrees with part of the multiplication table of a metric group. A recent result of Green shows that this metric approximation is necessary: it is not always possible to obtain a proportional-sized subset that agrees with part of the multiplication table of a group.Free products of digroupshttps://zbmath.org/1491.201872022-09-13T20:28:31.338867Z"Zhang, Guangliang"https://zbmath.org/authors/?q=ai:zhang.guangliang"Chen, Yuqun"https://zbmath.org/authors/?q=ai:chen.yuqun"Zhang, Zerui"https://zbmath.org/authors/?q=ai:zhang.zeruiSummary: We construct the free products of arbitrary digroups, and thus we solve an open problem of Zhuchok.The probability of commuting subgroups in arbitrary lattices of subgroupshttps://zbmath.org/1491.201882022-09-13T20:28:31.338867Z"Muhie, Seid Kassaw"https://zbmath.org/authors/?q=ai:muhie.seid-kassaw"Russo, Francesco G."https://zbmath.org/authors/?q=ai:russo.francesco-giuseppeSummary: A finite group \(G\), in which two randomly chosen subgroups \(H\) and \(K\) commute, has been classified by Iwasawa in 1941. It is possible to define a probabilistic notion, which ``measures the distance'' of \(G\) from the groups of Iwasawa. Here we introduce the generalized subgroup commutativity degree \(\mathrm{gsd}(G)\) for two arbitrary sublattices \(\mathrm{S}(G)\) and \(\mathrm{T}(G)\) of the lattice of subgroups \(\mathrm{L}(G)\) of \(G\). Upper and lower bounds for \(\mathrm{gsd}(G)\) are shown and we study the behaviour of \(\mathrm{gsd}(G)\) with respect to subgroups and quotients, showing new numerical restrictions.Configuration Lie groupoids and orbifold braid groupshttps://zbmath.org/1491.220012022-09-13T20:28:31.338867Z"Roushon, S. K."https://zbmath.org/authors/?q=ai:roushon.sayed-kSummary: We propose two definitions of configuration Lie groupoids and in both the cases we prove a Fadell-Neuwirth type fibration theorem for a class of Lie groupoids. We show that this is the best possible extension, in the sense that, for the class of Lie groupoids corresponding to global quotient orbifolds with nonempty singular set, the fibration theorems do not hold. Secondly, we prove a short exact sequence of fundamental groups (called \textit{pure orbifold braid groups}) of one of the configuration Lie groupoids of the Lie groupoid corresponding to the punctured complex plane with cone points. This shows the possibility of a quasifibration type Fadell-Neuwirth theorem for Lie groupoids.
As consequences, first we see that the pure orbifold braid groups have poly-virtually free structure, which generalizes the classical braid group case. We also provide an explicit set of generators of the pure orbifold braid groups. Secondly, we prove that a class of affine and finite complex Artin groups are virtually poly-free, which partially answers the question if all Artin groups are virtually poly-free [\textit{M. Bestvina}, Geom. Topol. 3, 269--302 (1999; Zbl 0998.20034), Question 2]. Finally, combining this poly-virtually free structure and a recent result [\textit{M. Bestvina} et al., The Farrell-Jones conjecture for hyperbolic-by-cyclic groups. (2021; \url{arXiv:2105.13291}], we deduce the Farrell-Jones isomorphism conjecture for the above class of orbifold braid groups. This also implies the conjecture for the case of the affine Artin group of type \(\widetilde{D}_n\), which was left open in [the author's paper, Can. J. Math. 73, No. 4, 1153--1170 (2021; Zbl 1485.19003); erratum ibid. 74, No. 2, 602 (2022), Problem].\(C^*\)-simplicity of HNN extensions and groups acting on treeshttps://zbmath.org/1491.220022022-09-13T20:28:31.338867Z"Bryder, Rasmus Sylvester"https://zbmath.org/authors/?q=ai:bryder.rasmus-sylvester"Ivanov, Nikolay A."https://zbmath.org/authors/?q=ai:ivanov.nikolay-a"Omland, Tron"https://zbmath.org/authors/?q=ai:omland.tron-aSummary: We study non-ascending HNN extensions acting on their Bass-Serre tree and characterize \(C^*\)-simplicity and the unique trace property by means of the kernel and quasi-kernels of the HNN extension in question. We also present a concrete example of an HNN extension that is a new example of a group that is not \(C^*\)-simple but does have the unique trace property. Additionally, we include certain more general results, mostly based on previous work of various authors, concerning \(C^*\)-simplicity of groups admitting extreme boundary actions, and in particular, groups acting on trees.Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic grouphttps://zbmath.org/1491.370102022-09-13T20:28:31.338867Z"Barbieri, Sebastián"https://zbmath.org/authors/?q=ai:barbieri.sebastian"García-Ramos, Felipe"https://zbmath.org/authors/?q=ai:garcia-ramos.felipe"Li, Hanfeng"https://zbmath.org/authors/?q=ai:li.hanfengSummary: We provide a unifying approach which links results on algebraic actions by \textit{D. Lind} and \textit{K. Schmidt} [J. Am. Math. Soc. 12, No. 4, 953--980 (1999; Zbl 0940.22004); Russ. Math. Surv. 70, No. 4, 657--714 (2015; Zbl 1357.37041); translation from Usp. Mat. Nauk 70, No. 4, 77--142 (2015)], \textit{N.-P. Chung} and \textit{H. Li} [Invent. Math. 199, No. 3, 805--858 (2015; Zbl 1320.37009)], and a topological result by \textit{T. Meyerovitch} [Ergodic Theory Dyn. Syst. 39, No. 9, 2570--2591 (2019; Zbl 1431.37030)]
that relates entropy to the set of asymptotic pairs. In order to do this we introduce a series of Markovian properties and, under the assumption that they are satisfied, we prove several results that relate topological entropy and asymptotic pairs (the homoclinic group in the algebraic case). As new applications of our method, we give a characterization of the homoclinic group of any finitely presented expansive algebraic action of (1) any elementary amenable group with an upper bound on the orders of finite subgroups or (2) any left orderable amenable group, using the language of independence entropy pairs.Stability in a grouphttps://zbmath.org/1491.370122022-09-13T20:28:31.338867Z"Conant, Gabriel"https://zbmath.org/authors/?q=ai:conant.gabrielSummary: We develop local stable group theory directly from topological dynamics, and extend the main tools in this subject to the setting of stability ``in a model.'' Specifically, given a group \(G\), we analyze the structure of sets \(A \subseteq G\) such that the bipartite relation \(xy\in A\) omits infinite half-graphs. Our proofs rely on the characterization of model-theoretic stability via Grothendieck's ``double-limit'' theorem (as shown by \textit{I. Ben Yaacov} [Bull. Symb. Log. 20, No. 4, 491--496 (2014; Zbl 1345.03058)]), and the work of \textit{R. Ellis} and \textit{M. Nerurkar} [Trans. Am. Math. Soc. 313, No. 1, 103--119 (1989; Zbl 0674.54026)]
on weakly almost periodic \(G\)-flows.Johnson pseudo-contractibility of certain semigroup algebras. IIhttps://zbmath.org/1491.430012022-09-13T20:28:31.338867Z"Askari-Sayah, M."https://zbmath.org/authors/?q=ai:askari-sayah.m"Pourabbas, A."https://zbmath.org/authors/?q=ai:pourabbas.abdolrasoul"Sahami, A."https://zbmath.org/authors/?q=ai:sahami.amirSummary: We investigate Johnson pseudo-contractibility and pseudo-contractibility of Clifford semigroup algebras. We show that, for a Clifford semigroup \(S\), if \(\ell^1(S)\) has a central approximate identity in \(c_{00}(S) \), then \(\ell^1(S)\) is (Johnson) pseudo-contractible if and only if \(E(S)\) is locally finite and each maximal subgroup of \(S\) is (amenable) finite, respectively. As an application, we characterize Johnson pseudo-contractibility and pseudo-contractibility of \(\ell^1(S) \), where \(S\) is a commutative semigroup, a band semigroup, or an inverse semigroup with totally ordered idempotents set.
For Part I, see [the second and third author, Semigroup Forum 97, No.~2, 203--213 (2018; Zbl 1429.43001)].On multipliers and compact multipliers of commutative semigroup algebra \(\ell^p(S,\omega)\)https://zbmath.org/1491.460432022-09-13T20:28:31.338867Z"Dabhi, Prakash A."https://zbmath.org/authors/?q=ai:dabhi.p-a"Likhada, Darshana B."https://zbmath.org/authors/?q=ai:likhada.darshana-bSummary: Let \(1<p<\infty\), and let \(q\) be the conjugate index of \(p\). Let \(\omega\) be a uniform weight on an abelian semigroup \(S\) satisfying \(\omega^{-q} * \omega^{-q}\leq \omega^{-q}\), and let \(\mathcal{A}\) be a commutative Banach algebra with approximate identity bounded by 1. We shall determine the multiplier algebra of \(\ell^p(S,\omega, \mathcal{A})\) under some conditions. We apply this result to characterize some compact multipliers on \(\ell^p(S,\omega)\) when \(S\) is cancellative. A necessary and sufficient condition for the semisimplicity of \(\ell^p(S,\omega)\) is also discussed.Boundary maps, germs and quasi-regular representationshttps://zbmath.org/1491.460512022-09-13T20:28:31.338867Z"Kalantar, Mehrdad"https://zbmath.org/authors/?q=ai:kalantar.mehrdad"Scarparo, Eduardo"https://zbmath.org/authors/?q=ai:scarparo.eduardo-pSummary: We investigate the tracial and ideal structures of \(C^\ast \)-algebras of quasi-regular representations of stabilizers of boundary actions. Our main tool is the notion of boundary maps, namely \(\Gamma \)-equivariant unital completely positive maps from \(\Gamma- C^\ast \)-algebras to \(C( \partial_F \Gamma)\), where \(\partial_F \Gamma\) denotes the Furstenberg boundary of a group \(\Gamma \). For a unitary representation \(\pi\) coming from the groupoid of germs of a boundary action, we show that there is a unique boundary map on \(C_\pi^\ast(\Gamma)\). Consequently, we completely describe the tracial structure of the \(C^\ast \)-algebras \(C_\pi^\ast(\Gamma)\), and for any \(\Gamma \)-boundary \(X\), we completely characterize the simplicity of the \(C^\ast \)-algebras generated by the quasi-regular representations \(\lambda_{\Gamma / \Gamma_x}\) associated to stabilizer subgroups \(\Gamma_x\) for any \(x \in X\). As an application, we show that the \(C^\ast \)-algebra generated by the quasi-regular representation \(\lambda_{T / F}\) associated to Thompson's groups \(F \leq T\) does not admit traces and is simple.Central amalgamation of groups and the RFD propertyhttps://zbmath.org/1491.460522022-09-13T20:28:31.338867Z"Shulman, Tatiana"https://zbmath.org/authors/?q=ai:shulman.tatianaSummary: It is an old and challenging topic to investigate for which discrete groups \(G\) the full group \(\mathrm{C}^*\)-algebra \(C^\ast(G)\) is residually finite-dimensional (RFD). In particular not much is known about how the RFD property behaves under fundamental constructions, such as amalgamated free products and HNN-extensions. In [\textit{K.~Courtney} and \textit{T.~Shulman}, Proc. Am. Math. Soc. 148, No.~2, 765--776 (2020; Zbl 1445.46043)] it was proved that central amalgamated free products of virtually abelian groups are RFD. In this paper we prove that this holds much beyond this case. Our method is based on showing a certain approximation property for characters induced from central subgroups. In particular it allows us to prove that free products of polycyclic-by-finite groups amalgamated over finitely generated central subgroups are RFD. On the other hand we prove that the class of RFD \(\mathrm{C}^*\)-algebras (and groups) is not closed under central amalgamated free products. Namely we give an example of RFD groups (in fact finitely generated amenable RF groups) whose central amalgamated free product is not RFD, moreover it is not even maximally almost periodic. This answers a question of
\textit{M.~S. Khan} and \textit{S.~A. Morris} [Trans. Am. Math. Soc. 273, 405--416, 417--432 (1982; Zbl 0496.22004)].On intermediate subalgebras of \(C^*\)-simple group actionshttps://zbmath.org/1491.460642022-09-13T20:28:31.338867Z"Amrutam, Tattwamasi"https://zbmath.org/authors/?q=ai:amrutam.tattwamasiSummary: We show that for a large class of actions \(\Gamma\curvearrowright\mathcal{A}\) of \(C^*\)-simple groups \(\Gamma\) on unital \(C^*\)-algebras \(\mathcal{A}\), including any non-faithful action of a hyperbolic group with trivial amenable radical, every intermediate \(C^*\)-subalgebra \(\mathcal{B}\), \(C_{\lambda}^*(\Gamma)\subseteq\mathcal{B}\subseteq\mathcal{A}\rtimes_r\Gamma\), is of the form \(\mathcal{A}_1\rtimes_r\Gamma\), where \(\mathcal{A}_1\) is a unital \(\Gamma -C^*\)-subalgebra of \(\mathcal{A}\).An infinite \(\{3,7\}\)-surfacehttps://zbmath.org/1491.510142022-09-13T20:28:31.338867Z"Lee, Dami"https://zbmath.org/authors/?q=ai:lee.damiBy constructive means, in this richly illustrated paper, with colored figures, the author proves that ``There exists an infinite \(\{3, 7\}\)-surface immersed in \({\mathbb R}^3\) that is a cover of Klein quartic.''
Reviewer: Victor V. Pambuccian (Glendale)The probability of spanning a classical space by two non-degenerate subspaces of complementary dimensionshttps://zbmath.org/1491.510172022-09-13T20:28:31.338867Z"Glasby, S. P."https://zbmath.org/authors/?q=ai:glasby.stephen-peter"Niemeyer, Alice C."https://zbmath.org/authors/?q=ai:niemeyer.alice-c"Praeger, Cheryl E."https://zbmath.org/authors/?q=ai:praeger.cheryl-eLet \(n,n^{\prime}\) be positive integers and let \(V\) be an (\(n + n^{\prime})\)-dimensional vector space over a finite field \(\mathbb{F}\) equipped with a nondegenerate alternating, hermitian or quadratic form. The authors estimate the proportion of pairs \((U, U^{\prime} )\), where \(U\) is a nondegenerate \(n\)-subspace and \(U^{\prime}\) is a nondegenerate \(n^{\prime}\)-subspace of \(V\), such that \(U +U^{\prime} = V\). The proportion is shown to be at least \(1 - c/|\mathbb{F}|\) for some constant \(c < 2\) in the symplectic or unitary cases, and \(c < 3\) in the orthogonal case.
The authors point out that the problem they are dealing with has connections to computations in finite classical groups.
Reviewer: Erich W. Ellers (Toronto)Pattern equivariant mass transport in aperiodic tilings and cohomologyhttps://zbmath.org/1491.520162022-09-13T20:28:31.338867Z"Kelly, Michael"https://zbmath.org/authors/?q=ai:kelly.michael"Sadun, Lorenzo"https://zbmath.org/authors/?q=ai:sadun.lorenzo-aThe starting point for this topic is the following issue: Given two countable and uniformly discrete point sets \(X_1\) and \(X_2\) in \(\mathbb{R}^n\), does there exist a bijection \(b : X_1 \rightarrow X_2\) such that the distance from points \(x \in X_1\) to corresponding points \(b(x)\in X_2\) is uniformly bounded? The existence of bounded transport is a consequence of the so-called Hall Marriage Theorem and the proof of the Schröder-Bernstein theorem. A possible generalization of the above question is where the discrete point sets \(X_1\) and \(X_2\) are replaced by locally finite measures \(\mu_1\) and \(\mu_2\).
In this paper the authors consider measures that always either be pure point or absolutely continuous, that is, given by point patterns or by density functions. They impose restrictions on the point patterns \(X_1\) and \(X_2\), or the continuous distributions \(f_1\) and \(f_2\), to be derivable from the pattern of an underlying tiling \(T\). If a repetitive and aperiodic tiling \(T\) of \(\mathbb{R}^n\) and two positive strongly pattern equivariant (PE) mass distributions \(f_1\) and \(f_2\) is given they ask the following:
\begin{itemize}
\item[1.] When does there exist a bounded transport from \(f_1\) to \(f_2\)?
\item[2.] When is it possible to do this transport in a weakly PE way?
\item[3.] When is it possible to do this transport in a strongly PE way?
\end{itemize}
The authors convert these questions to questions of cohomology. They associate a class \([f_i]\in \breve{H}^n (\Omega_T,\mathbb{R})\), the top real-valued Cech cohomology of the continuous hull \(\Omega_T\) of \(T\), to each mass distribution \(f_i\). The authors introduce subspaces of \(\breve{H}^n (\Omega_T,\mathbb{R})\) that are called ``asymptotically negligible'' and ``well-balanced'' classes.
The central results of the paper are summarized in the following theorems (using the notation of the authors):
Theorem: Let \([f_1]\) and \([f_2]\) be the classes in \(\breve{H}^n (\Omega_T,\mathbb{R})\) associated to the strongly PE mass distributions \(f_1\) and \(f_2\). Then,
\begin{itemize}
\item[1.] there exists a bounded transport from \(f_1\) to \(f_2\) if and only if \([f_1] - [f_2]\) is well balanced (WB);
\item[2.] there exists a weakly PE transport from \(f_1\) to \(f_2\) if and only if \([f_1] - [f_2]\) is asymptotically negligible (AN);
\item[3.] there exists a strongly PE transport from \(f_1\) to \(f_2\) if and only if \([f_1] = [f_2]\).
\end{itemize}
Theorem: Let \(T\) be a repetitive and aperiodic tiling of \(\mathbb{R}^n\). If
\begin{itemize}
\item[1.] \(n = 1\), or
\item[2.] \(T\) is a codimension-1 cut-and-project tiling with canonical window,
\end{itemize}
then every WB class in \(\breve{H}^n (\Omega_T,\mathbb{R})\) is AN. In particular, given strongly PE mass distributions \(f_1\) and \(f_2\), there exists a bounded transport from \(f_1\) to \(f_2\) if and only if there exists a strongly PE transport from \(f_1\) to \(f_2\).
Reviewer: Jenö Szirmai (Budapest)Kac-Moody symmetric spaceshttps://zbmath.org/1491.530682022-09-13T20:28:31.338867Z"Freyn, Walter"https://zbmath.org/authors/?q=ai:freyn.walter"Hartnick, Tobias"https://zbmath.org/authors/?q=ai:hartnick.tobias"Horn, Max"https://zbmath.org/authors/?q=ai:horn.max"Köhl, Ralf"https://zbmath.org/authors/?q=ai:kohl.ralfThe goal of this article of a foundational nature is to establish a general theory of Kac-Moody symmetric spaces over the field \(\mathbb{R}\). The non-Archimedian case is known under the name of ``masures'' or ``hovels'', discovered by \textit{S. Gaussent} and \textit{G. Rousseau} [Ann. Inst. Fourier 58, No. 7, 2605--2657 (2008; Zbl 1161.22007)]. The authors introduce the concept of \emph{topological} symmetric space which is a topological space endowed with symmetries centered at each of his points. These symmetries have to satisfy axioms that where highlighted by \textit{O. Loos} [Bull. Am. Math. Soc. 73, 250--253 (1967; Zbl 0149.41004); Math. Z. 99, 141--170 (1967; Zbl 0148.17403)]: a symmetry has its center as unique (local) fixed point, it is an involution, and there is a kind of associativity axiom. We denote by \(x \cdot y\) the image of \(y\) by the symmetry centered at \(x\).
First, the authors study a general topological symmetric space \(\mathcal X\) and in particular the notion of geodesics on \(\mathcal X\) (which are symmetric subspaces of \(\mathcal X\) which are homeorphic to the symmetric space \(\mathbb{R}\)). They show that one can associated a \(1\)-parameter subgroup of \(\Aut(\mathcal X)\) to any geodesic, and that it enjoys natural properties.
Second, a topological symmetric space is associated to any generalized Cartan matrix \(A\), which is assumed to be irreducible and symmetrizable (or equivalently to any irreducible symmetrizable real Kac-Moody group \(G\)). In the spherical case (when \(A\) corresponds to a finite-dimensional Lie group \(G\)), a symmetric space can be associated considering the set of conjugates of the Cartan involution of \(G\). For general \(A\), the authors consider the same definition, or rather a quotient of it by vector spaces of dimension the corank of \(A\). They define in this way a topological symmetric space denoted \(\overline{\mathcal X}_G\).
One important result is the study of the flats in the symmetric space \(\overline{\mathcal X}_G\). Flats \(F\) are here defined by three axioms: \(F\) is stable under symmetries, \(F\) contains an element \(z\) such that \(z \cdot x = y\) for any \(x,y \in F\), and the commutativity axiom \(x \cdot (z \cdot (y \cdot z)) = y \cdot (z \cdot (x \cdot z))\) holds for any \(x,y,z \in F\). It is shown that \(G\) acts transitively on the set of pairs \((p,F)\), where \(F\) is a flat in \(\overline{\mathcal X}_G\) and \(p \in F\). It follows that all flats are Euclidean of dimension the rank of \(A\). Moreover, the stabilizer in \(G\) of such a pair \((p,F)\) acts on \(F\) and its action is identified with the action of the Weyl group of \(G\).
Geodesic connectedness is also considered: all symmetric spaces \(\overline{\mathcal X}_G\) are geodesically connected, but if \(A\) is not spherical then two general points in \(\overline{\mathcal X}_G\) cannot be connected by only one geodesic.
In the non-spherical and non-affine case, the Tits cone is a non-trivial cone in the Cartan algebra. A global version of this cone is constructed by the authors and named causal structure. It associates to each point \(x\) in \(\overline{\mathcal X}_G\) a subset of \(\overline{\mathcal X}_G\) that intersects any flat containing \(x\) along a cone with tip \(x\), and that is invariant under the automorphism group. This causal structure allows defining the causal boundary, and it is shown among other things that an element in \(G\) is determined by its action on the causal boundary. The causal pre-order it defines is also studied.
Reviewer: Pierre-Emmanuel Chaput (Nancy)Archipelago groups are locally free. Corrigendum to: ``Cotorsion and wild homology''https://zbmath.org/1491.550132022-09-13T20:28:31.338867Z"Herfort, Wolfgang"https://zbmath.org/authors/?q=ai:herfort.wolfgang-n"Hojka, Wolfram"https://zbmath.org/authors/?q=ai:hojka.wolframSummary: An Archipelago group is the quotient of the topologist's product \(G = \circledast_{i \ge 1}G_i\) of a sequence \((G_i)_{i \ge 1}\) of groups modulo the normal closure of the subset \(\bigcup_{i \ge 1}G_i\) in \(G\). In this note we provide a simple proof of a result from [the authors, ibid. 221, No. 1, 275--290 (2017; Zbl 1421.55012)], namely that Archipelago groups are locally free.Conway's potential function via the Gassner representationhttps://zbmath.org/1491.570022022-09-13T20:28:31.338867Z"Conway, Anthony"https://zbmath.org/authors/?q=ai:conway.anthony"Estier, Solenn"https://zbmath.org/authors/?q=ai:estier.solennSummary: We show how Conway's multivariable potential function can be constructed using braids and the reduced Gassner representation. The resulting formula is a multivariable generalization of a construction, due to Kassel-Turaev, of the Alexander-Conway polynomial in terms of the Burau representation. Apart from providing an efficient method of computing the potential function, our result also removes the sign ambiguity in the current formulas which relate the multivariable Alexander polynomial to the reduced Gassner representation. We also relate the distinct definitions of this representation which have appeared in the literature.Slope of orderable Dehn filling of two-bridge knotshttps://zbmath.org/1491.570052022-09-13T20:28:31.338867Z"Gao, Xinghua"https://zbmath.org/authors/?q=ai:gao.xinghua|gao.xinghua.1|gao.xinghua.2For genus one, double twist knots, the interval consisting of slopes which yield closed \(3\)-manifolds with left-orderable fundamental groups is known from [\textit{R. Hakamata} and \textit{M. Teragaito}, Algebr. Geom. Topol. 14, No. 4, 2125--2148 (2014; Zbl 1311.57011)] and [\textit{A. T. Tran}, J. Math. Soc. Japan 67, No. 1, 319--338 (2015; Zbl 1419.57028)].
The purpose of the present paper is to give such an interval for the class of double twist knots with higher genus. (Parts of this result have also been proved by A. Tran.)
The main argument goes as follows. Examine real roots of the Riley polynomial of the knot, and construct a non-trivial representation of the knot group into the universal covering group of \(\mathrm{SL}_2(\mathbb{R})\), which is known to be left-orderable. Consider the translation extension locus introduced by \textit{M. Culler} and \textit{N. Dunfield} [Geom. Topol. 22, No. 3, 1405--1457 (2018; Zbl 1392.57012)] and the holonomy extension locus introduced by the author of the present paper [``Orderability of homology spheres obtained by Dehn filling'', Preprint, \url{arXiv:1810.11202}]. They are locally finite unions of analytic arcs and isolated points in \(H^1(\partial E(K);\mathbb{R})\cong \mathbb{R}^2\), where \(E(K)\) is the knot exterior. For a slope \(r\), if the line \(L_r\) through the origin with slope \(-r\) intersects the extension locus at a nonzero point which is neither parabolic nor ideal, then \(r\)-surgery yields a \(3\)-manifold whose fundamental group is left-orderable. A similar result holds for the holonomy extension locus.
The last part of the paper contains various examples of the translation extension locus and holonomy extension locus of some double twist knots and other \(2\)-bridge knots, made by a computer program. Based on these, a conjecture for general \(2\)-bridge knots is proposed.
Reviewer: Masakazu Teragaito (Hiroshima)A unified FFT-based approach to maximum assignment problems related to transitive finite group actionshttps://zbmath.org/1491.651762022-09-13T20:28:31.338867Z"Clausen, Michael"https://zbmath.org/authors/?q=ai:clausen.michaelSummary: This paper studies the cross-correlation of two real-valued functions whose common domain is a set on which a finite group acts transitively. Such a group action generalizes, e.g., circular time shifts for discrete periodic time signals or spatial translations in the context of image processing. Cross-correlations can be reformulated as convolutions in group algebras and are thus transformable into the spectral domain via Fast Fourier Transforms. The resulting general discrete cross-correlation theorem will serve as the basis for a unified FFT-based approach to maximum assignment problems related to transitive finite group actions. Our main focus is on those assignment problems arising from the actions of symmetric groups on the cosets of Young subgroups. The ``simplest'' nontrivial representatives of the latter class of problems are the NP-hard symmetric and asymmetric maximum quadratic assignment problem. We present a systematic spectral approach in terms of the representation theory of symmetric groups to solve such assignment problems. This generalizes and algorithmically improves Kondor's spectral branch-and-bound approach [\textit{R. Kondor}, in: Proceedings of the 21st annual ACM-SIAM symposium on discrete algorithms, SODA 2010, Austin, TX, USA, January 17--19, 2010. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM); New York, NY: Association for Computing Machinery (ACM). 1017--1028 (2010; Zbl 1288.68130)] to the exact solution of the asymmetric maximum quadratic assignment problem.Polynomial-time right-ideal morphisms and congruenceshttps://zbmath.org/1491.680772022-09-13T20:28:31.338867Z"Birget, J. C."https://zbmath.org/authors/?q=ai:birget.jean-camilleOn invariant subspaces in the Lai-Massey scheme and a primitivity reductionhttps://zbmath.org/1491.940332022-09-13T20:28:31.338867Z"Aragona, Riccardo"https://zbmath.org/authors/?q=ai:aragona.riccardo"Civino, Roberto"https://zbmath.org/authors/?q=ai:civino.robertoSummary: In symmetric cryptography, the round functions used as building blocks for iterated block ciphers are obtained as the composition of different layers acting as a sequence of bijective transformations providing global increasing complexity. The study of the conditions on such layers which make the group generated by the round functions of a block cipher a primitive group has been addressed in the past years, both in the case of Substitution Permutation Networks and Feistel Networks, giving to block cipher designers the recipe to avoid the imprimitivity attack, which exploits the invariance of some subspaces during the encryption. In the case of Lai-Massey schemes, where both Substitution Permutation Network and Feistel Network features are combined, the resistance against imprimitivity attacks has been a long-standing open problem. In this paper we consider a generalization of such a scheme and we prove its resistance against the imprimitivity attack. Our solution is obtained as a consequence of a more general result in which the problem of proving the primitivity of a generalized Lai-Massey scheme is reduced to the simpler one of proving the primitivity of the group generated by the round functions of a strictly related Substitution Permutation Network. We prove how this implies a reduction in the computational cost of invariant-subspace search.Two general schemes of algebraic cryptographyhttps://zbmath.org/1491.940642022-09-13T20:28:31.338867Z"Roman'kov, Vitaly"https://zbmath.org/authors/?q=ai:romankov.vitaly-aSummary: In this paper, we introduce two general schemes of algebraic cryptography. We show that many of the systems and protocols considered in literature that use two-sided multiplications are specific cases of the first general scheme. In a similar way, we introduce the second general scheme that joins systems and protocols based on automorphisms or endomorphisms of algebraic systems. Also, we discuss possible applications of the membership search problem in algebraic cryptanalysis. We show how an efficient decidability of the underlined membership search problem for an algebraic system chosen as the platform can be applied to show a vulnerability of both schemes. Our attacks are based on the linear or on the nonlinear decomposition method, which complete each other. We give a couple of examples of systems and protocols known in the literature that use one of the two introduced schemes with their cryptanalysis. Mostly, these protocols simulate classical cryptographic schemes, such as Diffie-Hellman, Massey-Omura and ElGamal in algebraic setting. Furthermore, we show that, in many cases, one can break the schemes without solving the algorithmic problems on which the assumptions are based.