Recent zbMATH articles in MSC 06https://zbmath.org/atom/cc/062022-09-13T20:28:31.338867ZWerkzeugA 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.The elements of advanced mathematicshttps://zbmath.org/1491.030022022-09-13T20:28:31.338867Z"Krantz, Steven G."https://zbmath.org/authors/?q=ai:krantz.steven-georgePublisher's description: This book has enjoyed considerable use and appreciation during its first four editions. With hundreds of students having learned out of early editions, the author continues to find ways to modernize and maintain a unique presentation.
What sets the book apart is the excellent writing style, exposition, and unique and thorough sets of exercises. This edition offers a more instructive preface to assist instructors on developing the course they prefer. The prerequisites are more explicit and provide a roadmap for the course. Sample syllabi are included.
As would be expected in a fifth edition, the overall content and structure of the book are sound.
This new edition offers a more organized treatment of axiomatics. Throughout the book, there is a more careful and detailed treatment of the axioms of set theory. The rules of inference are more carefully elucidated.
Additional new features include:
\begin{itemize}
\item An emphasis on the art of proof.
\item Enhanced number theory chapter presents some easily accessible but still-unsolved problems. These include the Goldbach conjecture, the twin prime conjecture, and so forth.
\item The discussion of equivalence relations is revised to present reflexivity, symmetry, and transitivity before we define equivalence relations.
\item The discussion of the RSA cryptosystem in Chapter 8 is expanded.
\item The author introduces groups much earlier. Coverage of group theory, formerly in Chapter 11, has been moved up; this is an incisive example of an axiomatic theory.
\end{itemize}
Recognizing new ideas, the author has enhanced the overall presentation to create a fifth edition of this classic and widely-used textbook.
See the reviews of the first, second and third editions in [Zbl 0860.03001; Zbl 0988.03002; Zbl 1243.03001]. For the fourth edition see [Zbl 1375.03002].A unified approach to structural limits and limits of graphs with bounded tree-depthhttps://zbmath.org/1491.030042022-09-13T20:28:31.338867Z"Nešetřil, Jaroslav"https://zbmath.org/authors/?q=ai:nesetril.jaroslav"Ossona de Mendez, Patrice"https://zbmath.org/authors/?q=ai:ossona-de-mendez.patriceThis monograph sets up a framework for studying limits of infinite sequences of finite relational structures, such as graphs, by associating, when possible, a ``limit object'', called modeling, to such a sequence. This framework generalizes the frameworks considered by graph theorists when studying the notions of graphon and graphing. To say something about the main results we need to know something about three central concepts of this framework.
A sequence of finite structures \((A_n)_{n \in \mathbb{N}}\) (having the same finite relational signature) is FO-convergent if for every first-order formula \(\varphi(x_1, \ldots, x_k)\), the proportion of \(k\)-tuples in \(A_n\) which satisfy \(\varphi\) converges as \(n\) tends to infinity.
A modeling is a relational structure with the additional features that its domain is a Borel space equipped with a probability measure and every first-order definable relation is measurable in the corresponding product \(\sigma\)-algebra.
A modeling \(M\) is called a modeling FO-limit of an FO-convergent sequence of finite structures \((A_n)_{n \in \mathbb{N}}\) if for every first-order formula the proportion of tuples in \(A_n\) which satisfy it converges to the probability (according to \(M\)) of the relation defined by that formula in \(M\).
The main definitions and results are collected in the first section of the monograph. The first of these states that if \(\mathcal{C}\) is a monotone class of finite graphs such that every \(FO\)-convergent sequence of graphs from \(\mathcal{C}\) has a modeling FO-limit, then \(\mathcal{C}\) is nowhere dense. It follows that there are FO-convergent sequences of graphs without a modeling FO-limit. It is conjectured that the converse implication also holds and Section 5.2 in the appendix reports on recent progress towards verifying the conjecture.
The remaining main results are concerned with (colored) graphs: (a) Every FO-convergent sequence of finite graphs with a fixed maximum degree has a modeling FO-limit, and (b) every FO-convergent sequence of finite colored trees with a fixed maximum height has a modeling FO-limit. The statement (b) is generalized (see Theorem 4.3.6), via the use of interpretations, to say roughly that (c) every FO-convergent sequence of finite colored graphs with fixed maximum tree height has a modeling FO-limit. Conjecture 1.3 proposes conditions under which a modeling should be a modeling FO-limit of an FO-convergent sequence of finite graphs of with fixed maximum degree. In the case of the statements (b) and (c), this monograph proves converses of the implications. More precisely: If \(M\) is a modeling such that its underlying first-order structure is a rooted colored tree with finite height and if \(M\) satisfies the Finitary Mass Transport Principle (FMTP) (Definition 4.20), then \(M\) is a modeling FO-limit of a sequence of finite rooted colored trees. Via the use of interpretations this result, which is technically hard to prove, can be generalized (in Theorem 4.3.6) to roughly the following: If \(M\) is a modeling that can be interpreted in a modeling \(T\) with the FMTP such that the underlying first-order structure of \(T\) is a rooted colored tree, then \(M\) is the modeling FO-limit of a sequence of finite graphs with a fixed bound on the tree height.
Section 2 develops general methods (for possible future use) involving (Lindenbaum-Tarski) Boolean algebras, Stone spaces, Ehrenfeucht-Fraisse games, the Gaifmain locality theorem and the notion of an interpretation of one structure in another. Section 3 studies modelings and their relationship to the concepts of Vapnik-Chervonenkis dimension, nowhere denseness, and random freeness. Section 4 studies sequences and limits of (colored) graphs with bounded tree-height, in particular rooted colored trees with bounded height, and the results mentioned above about such structures are proved here. The final Section 5 discusses open problems related to the theory developed in the monograph.
Reviewer: Vera Koponen (Uppsala)A reinterpretation of the semilattice semantics with applicationshttps://zbmath.org/1491.030192022-09-13T20:28:31.338867Z"Weiss, Yale"https://zbmath.org/authors/?q=ai:weiss.yaleThe goal of the paper is, using an idea of semilattice semantics introduced by \textit{A. Urquhart} [J. Symb. Log. 37, 159--169 (1972; Zbl 0245.02028)], for relevant logics, to construct semilattice semantics for intuitionistic propositional logic \(\mathbf{J}\) and Jankov's logic \(\mathbf{KC}\). In addition, semilattice semantics for (semi)-relevant logics \(\mathbf{SJ}\) and \(\mathbf{SKC}\) are presented.
A semilattice frame is a structure \(\mathfrak{F} = \langle S, 0, \sqcup \rangle\) where \(\langle S, \sqcup \rangle\) is a join-semilattice and \(0 \in S\) is lattice bottom element.
Let \(\mathcal{P}(S)\) be a power set of \(S\) and \(\Pi\) be a set of all propositional variables.
An intuitionistic semilattice model is a structure \(\mathfrak{M} = \langle \mathfrak{F},V \rangle\) where \(\mathfrak{F} = \langle S, 0, \sqcup \rangle\) is a semilattice frame and \(V :\Pi \cup \{\bot\} \longrightarrow \mathcal{P}(S)\) is subject to the following conditions:
\begin{itemize}
\item[1.] \(x \in V(p)\) implies \(x \sqcup y \in V(p)\)
\item[2.] \(x \in V(\bot)\) implies \(x \sqcup y \in V(\bot)\)
\item[3.] \(x \in V(\bot)\) implies \(x \in V(p)\) (for all \(p \in \Pi\))
\end{itemize}
The truth conditions for model \(\mathfrak{M}\) are defined as follows:
\begin{itemize}
\item[(a)] \(\models_x^\mathfrak{M} p\) if and only if \(x \in V(p)\)
\item[(b)] \(\models_x^\mathfrak{M} \bot\) if and only if \(x \in V(\bot)\)
\item[(c)] \(\models_x^\mathfrak{M} \varphi \land \psi\) if and only if \(\models_x^\mathfrak{M} \varphi\) and \(\models_x^\mathfrak{M} \psi\)
\item[(d)] \(\models_x^\mathfrak{M} \varphi \lor \psi\) if and only if \(\models_x^\mathfrak{M} \varphi\) or \(\models_x^\mathfrak{M} \psi\)
\item[(c)] \(\models_x^\mathfrak{M} \varphi \to \psi\) if and only if for all \(y \in S\), \(\not\models_y^\mathfrak{M} \varphi\) or \(\models_{x \sqcup y}^\mathfrak{M} \psi\)
\end{itemize}
As usual, formula \(\varphi\) is valid in model \(\mathfrak{M}\) if \(\models_0^\mathfrak{M} \varphi\). It is proven that semilattice models form a sound and complete semantics for \(\mathbf{J}\).
If we add to definition of model restriction \(x \in V(P)\) implies \(x \sqcup y \in V(p)\), we obtain a sound and complete semantics for Jankov's logic.
Reviewer: Alex Citkin (Warren)Risk analysis via Łukasiewicz logichttps://zbmath.org/1491.030252022-09-13T20:28:31.338867Z"Vitale, Gaetano"https://zbmath.org/authors/?q=ai:vitale.gaetanoSummary: In this paper, we apply logical methods to risk analysis. We study \textit{generalized events}, i.e. not yes-no events but \textit{continuous} ones. We define on this class of events a risk function and a measure over it to analyse risk in this context. We use Riesz MV-algebras as algebraic structures and their associated logic in support of our research, thanks to their relations with other applications. Moreover, we investigate on decidability of consequence problem for our class of risk.Investigation and corrigendum to some results related to \(g\)-soft equality and \(gf\)-soft equality relationshttps://zbmath.org/1491.030402022-09-13T20:28:31.338867Z"Al-Shami, Tareq M."https://zbmath.org/authors/?q=ai:al-shami.tareq-mohammedSummary: Since Molodtsov defined the concept of soft sets, many types of soft equality relations between two soft sets were discussed. Among these types are \(g\)-soft equality and \(gf\)-soft equality relations introduced in [\textit{M. Abbas} et al., ibid. 28, No. 6, 1191--1203 (2014; Zbl 1459.03086)] and [\textit{M. Abbas} et al., ibid. 31, No. 19, 5955--5964 (2017; Zbl 07459988)], respectively. In this paper, we first aim to show that some results obtained in [Zbl 1459.03086, loc. cit.; Zbl 07459988, loc. cit.] need not be true, by giving two counterexamples. Second, we investigate under what conditions these results are correct. Finally, we define and study the concepts of \(gf\)-soft union and \(gf\)-soft intersection for arbitrary family of soft sets.Fuzzy \(\alpha \)-cut and related mathematical structureshttps://zbmath.org/1491.030502022-09-13T20:28:31.338867Z"Jana, Purbita"https://zbmath.org/authors/?q=ai:jana.purbita"Chakraborty, Mihir K."https://zbmath.org/authors/?q=ai:chakraborty.mihir-kumarSummary: This paper deals with the notions called fuzzy \(\alpha \)-cut, fuzzy strict \(\alpha \)-cut and their properties. Algebraic structures arising out of the family of fuzzy \(\alpha \)-cuts and fuzzy strict \(\alpha \)-cuts have been investigated. Some significance and usefulness of fuzzy \(\alpha \)-cuts are discussed.Monadic classes of quantum B-algebrashttps://zbmath.org/1491.030822022-09-13T20:28:31.338867Z"Ciungu, Lavinia Corina"https://zbmath.org/authors/?q=ai:ciungu.lavinia-corinaSummary: The aim of this paper is to define the monadic quantum B-algebras and to investigate their properties. If the monadic operators are isotone, we show that they form a residuated pair. Special properties are studied for the particular case of monadic quantum B-algebras with pseudo-product, and a representation theorem for monadic quantum B-algebras with pseudo-product is proved. The monadic filters of monadic quantum B-algebras are defined, and their properties are studied. We prove that there is an isomorphism between the lattice of all filters of a monadic quantum B-algebra and the lattice of all filters of its subalgebra of fixed elements. Monadic operators on unital quantales are introduced, and the functional monadic quantale is constructed.Results on topological lattice effect algebrashttps://zbmath.org/1491.030832022-09-13T20:28:31.338867Z"Rakhshani, M. R."https://zbmath.org/authors/?q=ai:rakhshani.m-r"Rezaei, G. R."https://zbmath.org/authors/?q=ai:rezaei.g-r"Borzooei, R. A."https://zbmath.org/authors/?q=ai:borzooei.rajab-aliSummary: In this paper, we define the notion of topological lattice effect algebras and investigate some of their properties. By using Sasaki arrows and F-balls, we construct two topology on lattice effect algebras. Then we study separation axioms on lattice effect algebras. Specifically, we find some conditions under which a topological lattice algebra is a \(T_0,T_1\), and Hausdorff space. Finally, by using a strong filter and a quotient lattice effect algebra constructed by it, we investigate under what conditions this quotient lattice effect algebra will be a topological lattice effect algebra.Erratum to: ``Some result on simple hyper \(K\)-algebras''https://zbmath.org/1491.030862022-09-13T20:28:31.338867Z"Madadi-Dargahi, Soodabeh"https://zbmath.org/authors/?q=ai:madadi-dargahi.soodabeh"Nasr-Azadani, Mohammad Ali"https://zbmath.org/authors/?q=ai:nasr-azadani.mohammad-aliSummary: In this manuscript, we show that the Theorem 3.28 [\textit{T. Roudbari} and \textit{M. M. Zahedi}, ibid. 3, No. 2, 29--48 (2008; Zbl 1301.03068)] is not correct and modify it.State theory on bounded hyper EQ-algebrashttps://zbmath.org/1491.030872022-09-13T20:28:31.338867Z"Xin, Xiao Long"https://zbmath.org/authors/?q=ai:xin.xiaolongSummary: In a hyper structure \((X,\star)\), \(x\star y\) is a non-empty subset of \(X\). For a state \(s\), \(s(x\star y)\) need not be well defined. In this paper, by defining \(s^*(x\star y)=\sup\{s(z)\mid z\in x\star y\}\), we introduce notions of sup-Bosbach states, state-morphisms and sup-Riečan states on a bounded hyper EQ-algebra and discuss the related properties. The states on bounded hyper EQ-algebras are the generalization of states on EQ-algebras. Then we discuss the relations among sup-Bosbach states, state-morphisms and sup-Riečan states on bounded hyper EQ-algebras. By giving a counter example, we show that a sup-Bosbach state may not be a sup-Riečan state on a hyper EQ-algebra. We give conditions in which each sup-Bosbach state becomes a sup-Riečan state on bounded hyper EQ-algebras. Moreover, we introduce several kinds of congruences on bounded hyper EQ-algebras, by which we construct the quotient hyper EQ-algebras. By use of the state \(s\) on a bounded hyper EQ-algebra \(H\), we set up a state \(\bar{s}\) on the quotient hyper EQ-algebra \(H/\theta \). We also give the condition, by which a bounded hyper EQ-algebra admits a sup-Bosbach state.Zero-divisor graphs and total coloring conjecturehttps://zbmath.org/1491.051022022-09-13T20:28:31.338867Z"Khandekar, Nilesh"https://zbmath.org/authors/?q=ai:khandekar.nilesh"Joshi, Vinayak"https://zbmath.org/authors/?q=ai:joshi.vinayak-vSummary: In this paper, we prove that the zero-divisor graphs of a special class of pseudocomplemented posets satisfy the total coloring conjecture. Also, we determine the edge chromatic number of the zero-divisor graphs of this special class of pseudocomplemented posets. These results are applied to zero-divisor graphs of finite reduced commutative rings.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.Generalized matrix polynomials of tree Laplacians indexed by symmetric functions and the GTS posethttps://zbmath.org/1491.051912022-09-13T20:28:31.338867Z"Nagar, Mukesh Kumar"https://zbmath.org/authors/?q=ai:nagar.mukesh-kumar"Sivasubramanian, Sivaramakrishnan"https://zbmath.org/authors/?q=ai:sivasubramanian.sivaramakrishnanSummary: Let \(T\) be a tree on \(n\) vertices with Laplacian matrix \(L_T\) and \(q\)-Laplacian \(\mathcal{L}_T^q\). Let \(\texttt{GTS}_n\) be the generalized tree shift poset on the set of unlabelled trees on \(n\) vertices. Inequalities are known between coefficients of the immanantal polynomial of \(L_T\) and \(\mathcal{L}_T^q\) as one moves up the poset \(\texttt{GTS}_{n}\). Using the Frobenius characteristic, this can be thought as a result involving the Schur symmetric function \(s_\lambda \). In this paper, we use an arbitrary symmetric function to define a generalized matrix function of an \(n \times n\) matrix. When the symmetric function is the monomial and the forgotten symmetric function, we generalize such inequalities among coefficients of the generalized matrix polynomial of \(\mathcal{L}_T^q\) as one moves up the \(\texttt{GTS}_n\) poset.Rowmotion orbits of trapezoid posetshttps://zbmath.org/1491.052042022-09-13T20:28:31.338867Z"Dao, Quang Vu"https://zbmath.org/authors/?q=ai:dao.quang-vu"Wellman, Julian"https://zbmath.org/authors/?q=ai:wellman.julian"Yost-Wolff, Calvin"https://zbmath.org/authors/?q=ai:yost-wolff.calvin-l"Zhang, Sylvester W."https://zbmath.org/authors/?q=ai:zhang.sylvester-wSummary: Rowmotion is an invertible operator on the order ideals of a poset which has been extensively studied and is well understood for the rectangle poset. In this paper, we show that rowmotion is equivariant with respect to a bijection of \textit{Z. Hamaker} et al. [Int. Math. Res. Not. 2020, No. 2, 487--540 (2020; Zbl 1433.05041)] between order ideals of rectangle and trapezoid posets, thereby affirming a conjecture of \textit{S. Hopkins} [``Minuscule doppelgängers, the coincidental down-degree expectations property, and rowmotion'', Preprint, \url{arXiv:1902.07301}] that the rectangle and trapezoid posets have the same rowmotion orbit structures for order ideals. Our main tools in proving this are \(K\)-jeu-de-taquin and (weak) \(K\)-Knuth equivalence of increasing tableaux. We define almost minimal tableaux as a family of tableaux naturally arising from order ideals and show for any partition \(\lambda\), the almost minimal tableaux of shape \(\lambda\) are in different (weak) \(K\)-Knuth equivalence classes.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)Planar posets that are accessible from below have dimension at most 6https://zbmath.org/1491.060022022-09-13T20:28:31.338867Z"Biró, Csaba"https://zbmath.org/authors/?q=ai:biro.csaba"Bosek, Bartłomiej"https://zbmath.org/authors/?q=ai:bosek.bartlomiej"Smith, Heather C."https://zbmath.org/authors/?q=ai:smith.heather-c"Trotter, William T."https://zbmath.org/authors/?q=ai:trotter.william-t-jun"Wang, Ruidong"https://zbmath.org/authors/?q=ai:wang.ruidong"Young, Stephen J."https://zbmath.org/authors/?q=ai:young.stephen-jIn this paper, the authors consider dimension of posets. It is known that a planar poset of height \(h\) has dimension at most \(192h + 96\), while a planar poset with \(t\) minimal elements has dimension at most \(2t +1\). In particular, a planar poset with a unique minimal element has dimension at most 3.
Definition. Let \(\mathbb{D}\) be a plane diagram for a poset \(P\), and let \(x\) be a minimal element of \(P\). We say that \(x\) is accessible from below when there is a positive number \(\epsilon = \epsilon(x)\) so that any point \(p\) in the plane which is distinct from \(x\), on the vertical ray emanating downwards from \(x\) and within distance \(\epsilon\) from \(x\) is in the exterior region. In particular, a plane diagram \(\mathbb{D}\) is accessible from below if every minimal element of \(P\) is accessible from below.
In this paper, the authors extend the above stated result and prove the following result.
Theorem. If \(P\) is an accessible from below poset, then \(\dim(P ) \leq 6\).
Reviewer: S. K. Nimbhorkar (Aurangabad)Equivalent forms for a poset to be modular posethttps://zbmath.org/1491.060032022-09-13T20:28:31.338867Z"Kishore, T. Ravi"https://zbmath.org/authors/?q=ai:kishore.t-ravi"Sundarayya, P."https://zbmath.org/authors/?q=ai:sundarayya.pSummary: The notion of modular and distributive posets which generalize the corresponding notions from the lattice theory are introduced by \textit{J. Larmerová} and \textit{J. Rachůnek} [Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 27, 13--23 (1988; Zbl 0693.06003)]. Later some extended results of uniquely complemented lattice are derived to uniquely complemented posets. Now, in this paper, some equivalent conditions for a poset to be modular poset are given.Counting linear extensions of posets with determinants of hook lengthshttps://zbmath.org/1491.060042022-09-13T20:28:31.338867Z"Garver, Alexander"https://zbmath.org/authors/?q=ai:garver.alexander"Grosser, Stefan"https://zbmath.org/authors/?q=ai:grosser.stefan"Matherne, Jacob P."https://zbmath.org/authors/?q=ai:matherne.jacob-p"Morales, Alejandro"https://zbmath.org/authors/?q=ai:morales.alejandro-hResiduated operators in complemented posetshttps://zbmath.org/1491.060052022-09-13T20:28:31.338867Z"Chajda, Ivan"https://zbmath.org/authors/?q=ai:chajda.ivan"Länger, Helmut"https://zbmath.org/authors/?q=ai:langer.helmut-mThe orbit of closure-involution operations: the case of Boolean functionshttps://zbmath.org/1491.060062022-09-13T20:28:31.338867Z"Dassow, Jürgen"https://zbmath.org/authors/?q=ai:dassow.jurgenIn 1922, Kuratowski proved his ``closure-complement theorem'', which asserts that if \(T\) is a topological space and \(A\subseteq T\), then there are at most \(14\) different sets that can occur in the sequence \[ A, cA, kA, kcA, ckA, ckcA, kckA, \ldots, \] where \(kX=\overline{X}\) is topological closure and \(cX=T\setminus X\) is set-theoretic complementation. This paper concerns an analogue of Kuratowski's theorem.
Following Kuratowski, the focus is on a set \(T\), a closure operator \(k: {\mathcal P}(A)\to {\mathcal P}(A)\) and an involution on subsets \(c: {\mathcal P}(A)\to {\mathcal P}(A)\). The goal is to determine, given \(A\subseteq T\), how many sets can be generated by \(A\) using \(k\) and \(c\). One writes \(T(k,c)=\{n_1,n_2,\ldots\}\) for the set of possible outcomes as \(A\) ranges over subsets of \(T\). Thus, \(T(k,c)=\{1,3\}\) means that, given any subset \(A\subseteq T\), one will generate either \(1\) or \(3\) different sets using \(k\) and \(c\), and that both possibilities are realized.
In the first part of the paper, the topological space \(A\) of Kuratowski is replaced by the set \(\mathcal O\) of all Boolean functions, the topological closure operator \(k\) of Kuratowski is taken to be the superposition closure operator \(S\), and the set-theoretic complementation operator \(c\) of Kuratowski is taken to be one of the involutions from the set \(\{c, \neg, d\}\). Here \(c\) again denotes set-theoretic complementation, as in Kuratowski's theorem, but \(\neg\) is the negation operator (if \(A\subseteq {\mathcal O}\), then \(\neg A = \{\neg f\;|\;f\in A\}\)) and \(d\) is a conjugation by \(\neg\) (if \(A\subseteq {\mathcal O}\), then \(dA = \{\neg f(\neg x_1,\ldots,\neg x_n)\;|\;f(x_1,\ldots,x_n)\in A\}\)). The goal is to determine, given \(A\subseteq {\mathcal O}\), how many sets will be generated by \(A\) using \(S\) and one of the involutions \(i\in \{c,\neg,d\}\). The number of sets that can be generated depends on the choice of \(A\) and the choice of involution \(i\in \{c,\neg,d\}\). One writes \({\mathcal O}(S,i)=\{n_1,\ldots,n_r\}\) for the set of possible outcomes as \(A\) ranges over subsets of \({\mathcal O}\).
The results of the first part of the paper are:
\begin{itemize}
\item \({\mathcal O}(S,c) = \{2,4,6\}\).
\item \({\mathcal O}(S,d) = \{1,2,3,4\}\).
\item \({\mathcal O}(S,\neg) = \{1,2,3,4,5,7\}\) or \(\{1,2,3,4,5,6,7\}\). (The paper does not resolve whether \(6\in {\mathcal O}(k,\neg)\).)
\end{itemize}
The results of the second part of the paper involve closure operators \(c_1, c_2^n, c_3^n\) on \(\mathcal O\) that are different from the superposition operator \(S\). These closure operators, which have complicated definitions, are created for the purpose of showing that \({\mathcal O}(S,i)\) depends strongly on \(S\). The results obtained are that
\begin{itemize}
\item \({\mathcal O}(c_1,d) = \{1,2,3,\infty\}\).
\item \({\mathcal O}(c_2^n,d) = \{1,2,3\}\cup \{4,6,\ldots,2n+2\}\).
\item \({\mathcal O}(c_3^n,d) = \{1,2,3\}\cup \{5,7,\ldots,2n+3\}\).
\end{itemize}
The paper ends with a section suggesting problems and generalizations.
Reviewer: Keith Kearnes (Boulder)Some characterizations of pseudo-chains in pseudo-ordered setshttps://zbmath.org/1491.060072022-09-13T20:28:31.338867Z"Rai, Shashirekha B."https://zbmath.org/authors/?q=ai:rai.shashirekha-b"Bhatta, S. Parameshwara"https://zbmath.org/authors/?q=ai:bhatta.s-parameshwara\( \delta \)-primary ideals in latticeshttps://zbmath.org/1491.060082022-09-13T20:28:31.338867Z"Nimbhorkar, Shriram K."https://zbmath.org/authors/?q=ai:nimbhorkar.shriram-khanderao"Nehete, Jaya Y."https://zbmath.org/authors/?q=ai:nehete.jaya-ySummary: In this paper we define an ideal expansion on a lattice. Also we define a \(\delta \)-primary ideal and a weakly \(\delta \)-primary ideal with the help of an ideal expansion and investigate some properties of these ideals. We show that for an expansion function \(\delta \), an ideal is \(\delta \)-primary if and only if that ideal is strongly \(\delta \)-primary ideal of lattice. Also we define a \(\delta \)-twin-zero and prove some results based on \(\delta \)-twin-zero.Convexity on complete latticeshttps://zbmath.org/1491.060092022-09-13T20:28:31.338867Z"Liu, Hongping"https://zbmath.org/authors/?q=ai:liu.hongping"Shi, Fu-Gui"https://zbmath.org/authors/?q=ai:shi.fu-guiSummary: By means of closure systems and closure operators on complete lattices, a generalized convex structure under which classical convex structures and \(L\)-convex structures are consistent with each other is established. The related convex spaces and hull spaces are investigated, and it is shown that they are isomorphic to each other from the viewpoint of category. In order to further characterize this convex structure, the notion of enclosed order spaces and their corresponding mappings are introduced. It is proved that the category of enclosed order spaces is also isomorphic to that of convex spaces we presented.Topological structures in complete co-residuated latticeshttps://zbmath.org/1491.060102022-09-13T20:28:31.338867Z"Kim, Young-Hee"https://zbmath.org/authors/?q=ai:kim.young-hee-yun|kim.younghee"Kim, Yong Chan"https://zbmath.org/authors/?q=ai:kim.yong-chan|kim.yongchan|kim.yong-chan.1Summary: Information systems and decision rules with imprecision and uncertainty in data analysis are studied in complete residuated lattices. In this paper, we introduce the notions of Alexandrov pretopology (precotopology) and join-meet (meet-join) operators in complete co-residuated lattices. Moreover, their properties and examples are investigated.On amply g-supplemented latticeshttps://zbmath.org/1491.060112022-09-13T20:28:31.338867Z"Nebiyev, Celil"https://zbmath.org/authors/?q=ai:nebiyev.celil"Ökten, Hasan Hüseyin"https://zbmath.org/authors/?q=ai:okten.hasan-huseyinSummary: In this work, all lattices are complete modular lattices with the greatest element 1 and the smallest element 0. Let \(L\) be a lattice and \(a \in L\). If for every \(b \in L\) such that \(a \vee b = 1\), \(a\) has a g-supplement \(x\) in \(L\) such that \(x \leq b\), then we say \(a\) has ample g-supplements in \(L\). If every element of \(L\) has ample g-supplements in \(L\), then \(L\) is called an amply g-supplemented lattice (see [7]). In this work, some properties of these lattices are investigated.Applying the Czédli-Schmidt sequences to congruence properties of planar semimodular latticeshttps://zbmath.org/1491.060122022-09-13T20:28:31.338867Z"Grätzer, G."https://zbmath.org/authors/?q=ai:gratzer.georgeSummary: Following [\textit{G. Grätzer} and \textit{E. Knapp}, Acta Sci. Math. 73, No. 3--4, 445--462 (2007; Zbl 1223.06007); ibid. 74, No. 1--2, 37--47 (2008; Zbl 1164.06004); ibid. 75, No. 1--2, 29--48 (2009; Zbl 1199.06029); ibid. 76, No. 1--2, 3--26 (2010; Zbl 1224.06013)], a planar semimodular lattice \(L\) is rectangular, if the left boundary chain has exactly one doubly-irreducible element, \(c_l\), and the right boundary chain has exactly one doubly-irreducible element, \(c_r\), and these elements are complementary.\par The \textit{G. Czédli} and \textit{E. T. Schmidt} [Order 29, No. 3, 481--497 (2012; Zbl 1257.06005)] sequences, construct rectangular lattices. We use them to prove some structure theorems. In particular, we prove that for a slim (no \(\mathbf{M}_3\) sublattice) rectangular lattice \(L\), the congruence lattice \(\operatorname{Con}L\) has exactly \(\text{length}[c_l, 1]+ \text{length}[c_r, 1]\) dual atoms and a dual atom in \(\operatorname{Con}L\) is a congruence with exactly two classes. We also describe the prime ideals in a slim rectangular lattice.On residuation in paraorthomodular latticeshttps://zbmath.org/1491.060132022-09-13T20:28:31.338867Z"Chajda, I."https://zbmath.org/authors/?q=ai:chajda.ivan"Fazio, D."https://zbmath.org/authors/?q=ai:fazio.davideSummary: Paraorthomodular lattices are quantum structures of prominent importance within the framework of the logico-algebraic approach to (unsharp) quantum theory. However, at the present time it is not clear whether the above algebras may be regarded as the algebraic semantic of a logic in its own right. In this paper, we start the investigation of material implications in paraorthomodular lattices by showing that any bounded modular lattice with antitone involution \({\mathbf{A}}\) can be converted into a left-residuated groupoid if it satisfies a strengthened form of regularity. Moreover, the above condition turns out to be also necessary whenever \({\mathbf{A}}\) is distributive.Residuation in modular lattices and posetshttps://zbmath.org/1491.060142022-09-13T20:28:31.338867Z"Chajda, Ivan"https://zbmath.org/authors/?q=ai:chajda.ivan"Länger, Helmut"https://zbmath.org/authors/?q=ai:langer.helmut-mAn equational theory for \(\sigma \)-complete orthomodular latticeshttps://zbmath.org/1491.060152022-09-13T20:28:31.338867Z"Freytes, Hector"https://zbmath.org/authors/?q=ai:freytes.hectorSummary: The condition of \(\sigma \)-completeness related to orthomodular lattices places an important role in the study of quantum probability theory. In the framework of algebras with infinitary operations, an equational theory for the category of \(\sigma \)-complete orthomodular lattices is given. In this structure, we study the congruences theory and directly irreducible algebras establishing an equational completeness theorem. Finally, a Hilbert style calculus related to \(\sigma \)-complete orthomodular lattices is introduced and a completeness theorem is obtained.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.Orthomodular lattices as \(L\)-algebrashttps://zbmath.org/1491.060172022-09-13T20:28:31.338867Z"Wu, Yali"https://zbmath.org/authors/?q=ai:wu.yali"Yang, Yichuan"https://zbmath.org/authors/?q=ai:yang.yi-chuan|yang.yichuanSummary: We first prove that the axioms system of orthomodular \(L\)-algebra (\(O\)-\(L\)-algebras for short) as given in [\textit{W. Rump}, Forum Math. 30, No. 4, 973--995 (2018; Zbl 1443.06005)] are not independent by giving an independent axiom one. Then, two conditions for \textit{KL}-algebras to be Boolean are provided. Furthermore, some theorems of Holland are reproved using the self-similar closure of \(OM \)-\(L\)-algebras. In particular, the monoid operation of the self-similar closure is shown to be commutative.The \(r\)-signed Birkhoff transformhttps://zbmath.org/1491.060182022-09-13T20:28:31.338867Z"Ehrenborg, Richard"https://zbmath.org/authors/?q=ai:ehrenborg.richardGiven a graded poset, the authors define a so-called flag \(f\)-vector. Then the \(ab\)-index encodes that the flag \(f\)-vector of this poset as a non-commutative polynomial in the variables \(a\) and \(b\). Then they introduce the \(r\)-signed Birkhoff transform of a distributive lattice extending the known concept of the Birkhoff transform and show how to compute the \(ab\)-index of this \(r\)-signed Birkhoff transform from the \(ab\)-index of the distributive lattice. They obtain new expressions for the \(ab\)-index of the \(r\)-cubical lattice.
Reviewer: Ivan Chajda (Přerov)Ideals of core regular double Stone algebrahttps://zbmath.org/1491.060192022-09-13T20:28:31.338867Z"Srikanth, A. R. J."https://zbmath.org/authors/?q=ai:srikanth.ammu-r-j"Ravi Kumar, R. V. G."https://zbmath.org/authors/?q=ai:ravi-kumar.r-v-gA topological duality for \(k\times j\)-rough Heyting algebrashttps://zbmath.org/1491.060202022-09-13T20:28:31.338867Z"Almiñana, Federico"https://zbmath.org/authors/?q=ai:alminana.federico-gabriel"Pelaitay, Gustavo"https://zbmath.org/authors/?q=ai:pelaitay.gustavoSummary: \(k\)-rough Heyting algebras were introduced by Eric San Juan as an algebraic formalism for reasoning on finite increasing sequences over Boolean algebras in general and on generalizations of rough set concepts in particular. In this paper, \(k\times j\)-rough Heyting algebras are defined and investigated. These algebras constitute an extension of Heyting algebras and in \(j=2\) case they coincide with \(k\)-rough Heyting algebras. The aim of this paper is to give a topological study for these new class of algebras.Tense De Morgan \(S4\)-algebrashttps://zbmath.org/1491.060212022-09-13T20:28:31.338867Z"Segura, Cecilia"https://zbmath.org/authors/?q=ai:segura.ceciliaQuasi-pseudo-hoops: an extension to pseudo-hoopshttps://zbmath.org/1491.060222022-09-13T20:28:31.338867Z"Chen, Wenjuan"https://zbmath.org/authors/?q=ai:chen.wenjuan"Chen, Zhaoying"https://zbmath.org/authors/?q=ai:chen.zhaoying"Wang, Hongkai"https://zbmath.org/authors/?q=ai:wang.hongkaiSummary: In this paper, we introduce the notion of quasi-pseudo-hoops (\(qp\)-hoops, for short) as the generalization of pseudo-hoops. First we give some new notions in order to define \(qp\)-hoops. We investigate the basic properties of \(qp\)-hoops and also prove that any \(qp\)-hoop has the Riesz Decomposition Property. Second we discuss filters of \(qp\)-hoops and show that there exists a bijective correspondence between normal filters and filter congruences on any \(qp\)-hoop. Finally, we introduce and study some subclasses of \(qp\)-hoops. The subdirect product decomposition of a bounded \(qp\)-hoop is shown. We also present that bounded Wajsberg \(qp\)-hoops with additional conditions are equivalent to quasi-pseudo-MV algebras and bounded basic \(qp\)-hoops with additional conditions are equivalent to quasi-pseudo-BL algebras.An approach to stochastic processes via non-classical logichttps://zbmath.org/1491.060232022-09-13T20:28:31.338867Z"Di Nola, Antonio"https://zbmath.org/authors/?q=ai:di-nola.antonio"Dvurečenskij, Anatolij"https://zbmath.org/authors/?q=ai:dvurecenskij.anatolij"Lapenta, Serafina"https://zbmath.org/authors/?q=ai:lapenta.serafinaSummary: Within the infinitary variety of \(\sigma \)-complete Riesz MV-algebras \(\mathbf{RMV}_\sigma \), we introduce the algebraic analogue of a random variable as a homomorphism defined on the free algebra in \(\mathbf{RMV}_\sigma \). After a preliminary study of the proposed notion, we use it to define stochastic processes in the framework of non-classical logic (Łukasiewicz logic, more precisely) and we define stochastic independence.EMV-algebras -- extended MV-algebrashttps://zbmath.org/1491.060242022-09-13T20:28:31.338867Z"Dvurečenskij, Anatolij"https://zbmath.org/authors/?q=ai:dvurecenskij.anatolij"Zahiri, Omid"https://zbmath.org/authors/?q=ai:zahiri.omidSummary: The present paper is a survey on a new kind of algebras, called EMV-algebras, generalizing both MV-algebras and generalized Boolean algebras. The survey is based on our papers
[Fuzzy Sets Syst. 373, 116--148 (2019; Zbl 1423.06048); J. Aust. Math. Soc. 106, No. 2, 200--234 (2019; Zbl 1439.06007); Soft Comput. 22, No. 22, 7519--7537 (2018; Zbl 1401.06010); Soft Comput. 23, No. 17, 7513--7536 (2019; Zbl 1418.06008)]. For these algebras a top element is not assumed a priori. Every EMV-algebra can be covered by a system of MV-algebras where each of them uses the restriction of \(\oplus,\vee,\wedge\). We prove that every such an EMV-algebra without top element can be embedded into an EMV-algebra with top element as its maximal ideal, and every EMV-algebra with top element is termwise equivalent to an MV-algebra. We show that the classes of EMV-algebras are intimately connected with subvarieties of MV-algebras. We establish a categorical equivalence of the category of EMV-algebras without top element with a special category of MV-algebras. We will study states and state-morphisms, their topological properties, a Krein-Mil'man-type representation, and an integral representation. We show also a kind of the Loomis-Sikorski theorem. Finally, we present some results on free EMV-algebras.
For the entire collection see [Zbl 1459.06001].Locally \(\sigma \)-complete and locally complete EMV-algebrashttps://zbmath.org/1491.060252022-09-13T20:28:31.338867Z"Dvurečenskij, Anatolij"https://zbmath.org/authors/?q=ai:dvurecenskij.anatolij"Zahiri, Omid"https://zbmath.org/authors/?q=ai:zahiri.omidSummary: In the paper, we introduce the concept of a locally complete EMV-algebra and a locally \(\sigma \)-complete EMV-algebra. They are generalizations of a complete and \(\sigma \)-complete MV-algebra, respectively. We will find some representations for locally complete EMV-algebras with atomic idempotent elements. We prove that each EMV-algebra of this type is a subdirect product of directly indecomposable complete EMV-algebras with top element. Also, we find a necessary and sufficient condition for a locally complete EMV-algebra when its representing EMV-algebra with the greatest element 1 is complete. Finally, we investigate the Cantor-Bernstein-type theorems for locally \(\sigma \)-complete EMV-algebras and \(\sigma \)-complete EMV-algebras.A characterization of pseudofinite MV-algebrashttps://zbmath.org/1491.060262022-09-13T20:28:31.338867Z"Farsimadan, Eslam"https://zbmath.org/authors/?q=ai:farsimadan.eslam"Lenzi, Giacomo"https://zbmath.org/authors/?q=ai:lenzi.giacomo"Rizzo, Paolo"https://zbmath.org/authors/?q=ai:rizzo.paolo"Borumand Saeid, Arsham"https://zbmath.org/authors/?q=ai:borumand-saeid.arshamSummary: We consider pseudofinite MV-algebras. As a main result, we show that an infinite MV-algebra is pseudofinite if and only if it is definably well founded, improving a result of a previous paper. Moreover, we show that the theory of pseudofinite MV-algebras has a partial form of elimination of quantifiers. Further, we show that the class of pseudofinite MV-chains and the class of pseudofinite MV-algebras are not finitely axiomatizable, we give some collapsing results for pseudofinite MV-algebras, we consider relative subalgebras of pseudofinite MV-algebras, and we study ideals of pseudofinite MV-algebras.When is the space of continuous functions a hyperarchimedean MV-algebra?https://zbmath.org/1491.060272022-09-13T20:28:31.338867Z"Jordan, Francis"https://zbmath.org/authors/?q=ai:jordan.francis"Pajoohesh, Homeira"https://zbmath.org/authors/?q=ai:pajoohesh.homeiraSummary: In this article we consider \(C(X,[0,1])\) as an MV-algebra. We show that \(C(X,[0, 1])\) is hyperarchimedian if and only if \(X\) is a finite union of spaces that only admit constant real-valued continuous functions. We also characterize the essential ideals and implicative ideals of \(C(X,[0,1])\). Lastly, we characterize the molecules of \(C(X,[0, 1])\).The two-sorted algebraic theory of states, and the universal states of MV-algebrashttps://zbmath.org/1491.060282022-09-13T20:28:31.338867Z"Kroupa, Tomáš"https://zbmath.org/authors/?q=ai:kroupa.tomas"Marra, Vincenzo"https://zbmath.org/authors/?q=ai:marra.vincenzoSummary: States of unital abelian lattice-groups (normalised positive group homomorphisms to \(\mathbb{R})\) provide an abstraction of expected-value operators. A well-known theorem due to Mundici asserts that the category of unital lattice-groups (with unit-preserving lattice-group homomorphisms as morphisms) is equivalent to the algebraic category of MV-algebras, and their homomorphisms. Through this equivalence, states of lattice-groups correspond to certain \([0,1]\)-valued functionals on MV-algebras, which are also known as states. In this paper we allow states to take values in any unital lattice-group (or in any MV-algebra) rather than just in \(\mathbb{R}\) (or just in \([0,1]\), respectively). We introduce a two-sorted algebraic theory whose models are precisely states of MV-algebras. We extend Mundici's equivalence to one between the category of MV-algebras with states as morphisms, and the category of unital abelian lattice-groups with, again, states as morphisms. Thus, the models of our two-sorted theory may also be regarded as states between unital abelian lattice-groups. As our first main result, we derive the existence of the \textit{universal state} of any MV-algebra from the existence of free algebras in multi-sorted algebraic categories. The significance of the universal state of a given algebra is that it provides the most general expected-value operator on that algebra -- a construct that is not available if one insists that states be real-valued. In the remaining part of the paper, we seek concrete representations of such universal states. We begin by clarifying the relationship of universal states with the theory of affine representations: the universal state \(A\to B\) of the MV-algebra \(A\) coincides with a certain modification of Choquet's affine representation (of the lattice-group corresponding to \(A)\) if, and only if, \(B\) is semisimple. Locally finite MV-algebras are semisimple, and Boolean algebras are instances of locally finite MV-algebras. Our second main result is then that the universal state of any locally finite MV-algebra has semisimple codomain, and can thus be described through our adaptation of Choquet's affine representation.EMV-algebras with quantifier and semi-states on EMV-algebrashttps://zbmath.org/1491.060292022-09-13T20:28:31.338867Z"Liu, Hongxing"https://zbmath.org/authors/?q=ai:liu.hongxing|liu.hongxing.1Summary: In this paper, we introduce and study EMVQ-algebras. We give some properties of EMVQ-algebras and prove that every EMVQ-algebra without top element can be EMVQ-embedded into an EMVQ-algebra \((N_0,\exists_0)\) with top element as a maximal monadic ideal of \((N_0, \exists_0)\). Also, every semisimple EMVQ-algebra is isomorphic to an EMV-clan with quantifier. The properties of semi-states on EMV-algebras are studied and a representation theorem on semi-state morphisms between EMValgebras is given.EBL-algebrashttps://zbmath.org/1491.060302022-09-13T20:28:31.338867Z"Liu, Hongxing"https://zbmath.org/authors/?q=ai:liu.hongxing.1|liu.hongxingSummary: In this paper, we define the notion of EBL-algebras, which are generalizations of BL-algebras and EMV-algebras. The notions of ideals, congruences and filters in EBL-algebras are introduced, and their mutual relationships are investigated. There is a one-to-one correspondence between the set of all ideals in an EBL-algebra and the set of all congruences on an EBL-algebra. Moreover, we give a representation theorem on EBL-algebras. Every proper EBL-algebras under some condition can be embedded into an EBL-algebras with a top element as an ideal.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.Hyper commutative basic algebras, hyper MV-algebras, and stateshttps://zbmath.org/1491.060322022-09-13T20:28:31.338867Z"Wang, Jing"https://zbmath.org/authors/?q=ai:wang.jing.18"Wu, Yali"https://zbmath.org/authors/?q=ai:wu.yali"Yang, Yichuan"https://zbmath.org/authors/?q=ai:yang.yi-chuan|yang.yichuanSummary: In this paper, we introduce hyper commutative basic algebras. We first prove that every hyper MV-algebra is a hyper commutative basic algebra. Then we show that a hyper commutative basic algebra of cardinality 2 is a hyper MV-algebra, which is similar to the result of Botur and Halaš that finite commutative basic algebras coincide with MV-algebras. However, we find a hyper commutative basic algebra of cardinality 3 which is not a hyper MV-algebra. Finally, we study two types of states on hyper commutative basic algebras.Basic algebras and L-algebrashttps://zbmath.org/1491.060332022-09-13T20:28:31.338867Z"Wang, Jing"https://zbmath.org/authors/?q=ai:wang.jing.18"Wu, Yali"https://zbmath.org/authors/?q=ai:wu.yali"Yang, Yichuan"https://zbmath.org/authors/?q=ai:yang.yi-chuan|yang.yichuanSummary: In this paper, we study the relation between L-algebras and basic algebras. In particular, we construct a lattice-ordered effect algebra which improves an example of \textit{I. Chajda} et al. [Algebra Univers. 60, No. 1, 63--90 (2009; Zbl 1219.06013)].Ehoopshttps://zbmath.org/1491.060342022-09-13T20:28:31.338867Z"Xie, Fei"https://zbmath.org/authors/?q=ai:xie.fei|xie.fei.1"Liu, Hongxing"https://zbmath.org/authors/?q=ai:liu.hongxing.1|liu.hongxingSummary: In this paper, we introduce the notion of Ehoops, which are generalizations of hoops. Unlike the hoop, an Ehoop does not necessarily have a top element. The notions of ideals and filters in Ehoops are defined. In Ehoops, both ideals and filters can be used to define the congruences. It is proved that if an Ehoop A satisfies the double negation property, there is a one-to-one correspondence between the set of all ideals of A and the set of all congruences on A. Prime ideal theorem on Ehoops is also given. In addition, we define the notions of implicative filters and positive implicative filters of Ehoops and investigate the quotient algebras induced by (positive) implicative filters.A duality for two-sorted latticeshttps://zbmath.org/1491.060352022-09-13T20:28:31.338867Z"Rivieccio, Umberto"https://zbmath.org/authors/?q=ai:rivieccio.umberto"Jung, Achim"https://zbmath.org/authors/?q=ai:jung.achimSummary: A series of representation theorems (some of which discovered very recently) present an alternative view of many classes of algebras related to non-classical logics (e.g. bilattices, semi-De Morgan, Nelson and quasi-Nelson algebras) as two-sorted algebras in the sense of many-sorted universal algebra. In all the above-mentioned examples, we are in fact dealing with a pair of lattices related by two meet-preserving maps. We use this insight to develop a Priestley-style duality for such structures, mainly building on the duality for meet-semilattices of G. Bezhanishvili and R. Jansana. Our approach simplifies all the existing dualities for these algebras and is applicable more generally; in particular, we show how it specialises to the class of quasi-Nelson algebras, which has not yet been studied from a duality point of view.Intuitionistic fuzzy congruences on product latticeshttps://zbmath.org/1491.060362022-09-13T20:28:31.338867Z"Rasuli, Rasul"https://zbmath.org/authors/?q=ai:rasuli.rasulSummary: In this work, the concept of intuitionistic fuzzy congruences on lattice \(X\) was introduced and was defined direct product between them. Also some characterizations of them were established. Finally, isomorphism between factor lattices of similarity classes was investigated.Semidistributivity and whitman property in implication zroupoidshttps://zbmath.org/1491.060372022-09-13T20:28:31.338867Z"Cornejo, Juan M."https://zbmath.org/authors/?q=ai:cornejo.juan-manuel"Sankappanavar, Hanamantagouda P."https://zbmath.org/authors/?q=ai:sankappanavar.hanamantagouda-pSummary: In 2012, the second author [Sci. Math. Jpn. 75, No. 1, 21--50 (2012; Zbl 1279.06009)] introduced, and initiated the investigations into, the variety \(\mathcal{I}\) of implication zroupoids that generalize De Morgan algebras and \(\vee\)-semilattices with 0. An algebra \(\mathbf{A}=\langle A,\rightarrow,0\rangle\), where \(\rightarrow\) is binary and 0 is a constant, is called an \textit{implication zroupoid} (\(\mathcal{I}\)-zroupoid, for short) if \textbf{A} satisfies: \((x\rightarrow y)\rightarrow z\approx [(z'\rightarrow x)\rightarrow (y\rightarrow z)']'\), where \(x':=x\rightarrow 0\), and \(0''\approx 0\). Let \(\mathcal{I}\) denote the variety of implication zroupoids and \(\mathbf{A}\in\mathcal{I}\). For \(x,y\in\mathbf{A}\), let \(x\wedge y:=(x\rightarrow y')'\) and \(x\vee y:=((x'\wedge y')'\). In an earlier paper, we had proved that if \(\mathbf{A}\in\mathcal{I}\), then the algebra \(\mathbf{A}_{mj}=\langle A,\vee,\wedge\rangle\) is a bisemigroup. The purpose of this paper is two-fold: First, we generalize the notion of semidistributivity from lattices to bisemigroups and prove that, for every \(\mathbf{A}\in\mathcal{I}\), the bisemigroup \(\mathbf{A}_{mj}\) is semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety \(\mathcal{MEJ}\) of \(\mathcal{I}\), defined by the identity: \(x\wedge y \approx x\vee y\), satisfies the Whitman Property. We conclude the paper with two open problems.Closure prime spectrum of \(MS\)-almost distributive latticeshttps://zbmath.org/1491.060382022-09-13T20:28:31.338867Z"Rafi, N."https://zbmath.org/authors/?q=ai:rafi.noorbhasha"Bandaru, Ravi Kumar"https://zbmath.org/authors/?q=ai:bandaru.ravikumar|bandaru.ravi-kumar"Srujana, M."https://zbmath.org/authors/?q=ai:srujana.mSummary: In this paper, the concept of closure ideal is introduced in an \(MS\)-ADL and their properties are studied. It is observed that the set of all closure ideals forms a De Morgan ADL and topological properties of prime closure ideals are studied in an \(MS\)-ADL. Finally, equivalent conditions are provided for prime closure ideal to become maximal.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-ahmadOrdered regular pointed semigroups with biggest associateshttps://zbmath.org/1491.060412022-09-13T20:28:31.338867Z"Pinto, G. A."https://zbmath.org/authors/?q=ai:pinto.guilherme-aSummary: We consider the class \(\mathbf{BA}\) of ordered regular semigroups in which each element has a biggest associate \(x^\dag = \max\{y\mid xyx = x\}\). We investigate those ordered regular semigroups with biggest associates that are \textit{pointed} in the sense, that the classes modulo Green's relations \(\mathcal{R}\), \(\mathcal{L}\), \(\mathcal{D}\) have biggest elements which, are idempotent. Such a semigroup is necessarily a semiband. If the semigroup is also naturally ordered then, it is principally ordered. Generalisations of results in pointed principally ordered regular semigroups are obtained: (1) description of the subalgebra of \((S,{}^\dag)\) generated by a pair of comparable idempotents that are \(\mathcal{D}\)-related; (2) those \(\mathcal{D}\)-classes which are subsemigroups, are ordered rectangular bands.On 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.Correction to: ``Injective hulls for ordered algebras''https://zbmath.org/1491.060442022-09-13T20:28:31.338867Z"Zhang, Xia"https://zbmath.org/authors/?q=ai:zhang.xia"Laan, Valdis"https://zbmath.org/authors/?q=ai:laan.valdis"Feng, Jianjun"https://zbmath.org/authors/?q=ai:feng.jianjun"Reimaa, Ülo"https://zbmath.org/authors/?q=ai:reimaa.uloSummary: We show that Lemma 4.4 and Theorem 4.5 in [the first author and the second author, ibid. 76, No. 3, 339--349 (2016; Zbl 1404.06015)] are incorrect. These results can be corrected by replacing unary polynomial functions by linear functions.On quasi-commutative KK-algebrahttps://zbmath.org/1491.060452022-09-13T20:28:31.338867Z"Asawasamrit, Suphawat"https://zbmath.org/authors/?q=ai:asawasamrit.suphawat"Promsakon, Chanon"https://zbmath.org/authors/?q=ai:promsakon.chanon(no abstract)Isomorphism theorems in generalized \(d\)-algebrashttps://zbmath.org/1491.060462022-09-13T20:28:31.338867Z"Chaudhry, Muhammad Anwar"https://zbmath.org/authors/?q=ai:chaudhry.muhammad-anwar"Qureshi, Muhammad Imran"https://zbmath.org/authors/?q=ai:qureshi.muhammad-imran.1|qureshi.muhammad-imran"Fahad, Asfand"https://zbmath.org/authors/?q=ai:fahad.asfand"Bashir, Muhammad Sajjad"https://zbmath.org/authors/?q=ai:bashir.muhammad-sajjad(no abstract)On nayo algebrashttps://zbmath.org/1491.060472022-09-13T20:28:31.338867Z"Ilojide, Emmanuel"https://zbmath.org/authors/?q=ai:ilojide.emmanuelSummary: In this paper, nayo algebras are introduced. Properties of homomorphisms in relation to translation maps in nayo algebras are investigated. Moreover, monics and krib maps are introduced and studied in some classes of nayo algebras.\(t\)-derivations on complicated subtraction algebrashttps://zbmath.org/1491.060482022-09-13T20:28:31.338867Z"Jana, Chiranjibe"https://zbmath.org/authors/?q=ai:jana.chiranjibe"Senapati, Tapan"https://zbmath.org/authors/?q=ai:senapati.tapan"Pal, Madhumangal"https://zbmath.org/authors/?q=ai:pal.madhumangal(no abstract)Monadic RM algebrashttps://zbmath.org/1491.060492022-09-13T20:28:31.338867Z"Jastrzębska, Małgorzata"https://zbmath.org/authors/?q=ai:jastrzebska.malgorzata"Walendziak, Andrzej"https://zbmath.org/authors/?q=ai:walendziak.andrzejSummary: In this paper we introduce and study the monadic RM algebras. As particular cases we get monadic *RM algebras, monadic RM** algebras, monadic BE algebras, monadic BCI algebras, monadic BCK algebras, etc. The monadic deductive systems and R-congruence relations of a monadic RM algebra are defined and their properties are investigated. For the monadic *RM** algebras, it is proved that there is a one-to-one order-preserving correspondence between the R-congruence relations and the monadic closed t-deductive systems. Finally, we prove that if \(\mathcal{A}\) is a monadic pre-BBBZ algebra, then the lattice of monadic deductive systems of \(\mathcal{A}\) is isomorphic to the lattice of the deductive systems of \((A_{\exists\forall},\rightarrow ,1)\), where \(A_{\exists\forall}\) is the set of all fixed elements of \(\mathcal{A}\).Applications of fuzzy \(\rho \)-ideals in \(\rho \)-algebrashttps://zbmath.org/1491.060502022-09-13T20:28:31.338867Z"Khalil, Shuker Mahmood"https://zbmath.org/authors/?q=ai:khalil.shuker-mahmood"Hameed, Fatima"https://zbmath.org/authors/?q=ai:hameed.fatimaSummary: Some new notions of fuzzy algebras like fuzzy \(\rho \)-subalgebra, fuzzy \(\rho \)-ideal, and fuzzy \(\overline{\rho} \)-ideal are introduced in this work. Moreover, the relationships between our new notions and other types of fuzzy algebras like fuzzy \(d\)-subalgebra, fuzzy \(d\)-ideal, fuzzy BCK-subalgebra, and fuzzy BCK-ideal are investigated. Also, some basic characterizations of fuzzy \(\rho \)-ideal with their applications on images, Cartesian product, upper level, characteristic function, and strongest fuzzy relation are studied and discussed in this paper. Furthermore, several examples are presented to expound our notions in this work.Fuzzy set theoretic approach to generalized ideals in BCK/BCI-algebrashttps://zbmath.org/1491.060512022-09-13T20:28:31.338867Z"Muhiuddin, G."https://zbmath.org/authors/?q=ai:muhiuddin.ghulam"Alam, N."https://zbmath.org/authors/?q=ai:alam.noor"Obeidat, S."https://zbmath.org/authors/?q=ai:obeidat.sofian"Khan, N. M."https://zbmath.org/authors/?q=ai:khan.noor-mohammad"Zaidi, H. N."https://zbmath.org/authors/?q=ai:zaidi.hasan-nihal"Kirmani, S. A. K."https://zbmath.org/authors/?q=ai:kirmani.s-a-k"Altaleb, A."https://zbmath.org/authors/?q=ai:altaleb.anas"Aqib, J. M."https://zbmath.org/authors/?q=ai:aqib.j-m(no abstract)An overview of cubic intuitionistic \(\beta\)-subalgebrashttps://zbmath.org/1491.060522022-09-13T20:28:31.338867Z"Muralikrishna, P."https://zbmath.org/authors/?q=ai:muralikrishna.prakasam"Borumand Saeid, A."https://zbmath.org/authors/?q=ai:borumand-saeid.arsham"Vinodkumar, R."https://zbmath.org/authors/?q=ai:vinodkumar.r"Palani, G."https://zbmath.org/authors/?q=ai:palani.g-sThe basic concepts of cubic intuitionistic sets are adopted to \(\beta\)-subalgebras. The obtained results are typical for this theory.
Reviewer: Wiesław A. Dudek (Wrocław)Construction of an HV-BE-algebra from a BE-algebra based on ``begins lemma''https://zbmath.org/1491.060532022-09-13T20:28:31.338867Z"Naghibi, R."https://zbmath.org/authors/?q=ai:naghibi.razieh"Anvariyeh, S. M."https://zbmath.org/authors/?q=ai:anvariyeh.said-m|anvariyeh.seid-mohammad"Mirvakili, S."https://zbmath.org/authors/?q=ai:mirvakili.saeedSummary: In this paper, first we introduce the new class of HV-BE-algebra as a generalization of a (hyper) BE-algebra and prove some basic results and present several examples. Then, we construct the HV-BE-algebra associated to a BE-algebra (namely BL-BE-algebra) based on ``Begins lemma'' and investigate it.Ideals of transitive BE-algebrashttps://zbmath.org/1491.060542022-09-13T20:28:31.338867Z"Prabhakar, M. Bala"https://zbmath.org/authors/?q=ai:prabhakar.m-bala"Vali, S. Kalesha"https://zbmath.org/authors/?q=ai:vali.s-kalesha"Rao, M. Sambasiva"https://zbmath.org/authors/?q=ai:rao.m-sambasiva|rao.mukkamala-sambasivaSummary: The notion of ideals is introduced in transitive BE-algebras. Some characterization theorems of ideals of transitive BE-algebras are derived. The notion of semi-ideals is introduced and studied a relationship between semi-ideals and ideals. Properties of ideals are studied with the help of homomorphisms and congruences.Generalized lower sets of transitive BE-algebrashttps://zbmath.org/1491.060552022-09-13T20:28:31.338867Z"Prabhakar, M. Bala"https://zbmath.org/authors/?q=ai:prabhakar.m-bala"Vali, S. Kalesha"https://zbmath.org/authors/?q=ai:vali.s-kalesha"Rao, M. Sambasiva"https://zbmath.org/authors/?q=ai:rao.m-sambasiva|rao.mukkamala-sambasivaSummary: The notion of generalized lower sets is introduced in transitive BE-algebras. Some properties of generalized lower sets are investigated in transitive BE-algebras. Furthermore, a sufficient condition is derived for every generalized lower set BE-algebra to become an ideal.A new extension of RM-algebrashttps://zbmath.org/1491.060562022-09-13T20:28:31.338867Z"Rezaei, Akbar"https://zbmath.org/authors/?q=ai:rezaei.akbar"Borumand Saeid, Arsham"https://zbmath.org/authors/?q=ai:borumand-saeid.arshamIsomorphism theorems on weak AB-algebrashttps://zbmath.org/1491.060572022-09-13T20:28:31.338867Z"Sriponpaew, Boonyong"https://zbmath.org/authors/?q=ai:sriponpaew.boonyong"Sassanapitax, Lee"https://zbmath.org/authors/?q=ai:sassanapitax.leeSummary: In this paper, we introduce the notion of weak AB-algebras, which are a generalization of BCC-algebras. The concepts of congruences and formation of quotients of these algebras are demonstrated. In addition, we prove the fundamental theorems of isomorphism for weak AB-algebras.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-abrahamMeet infinite distributivity for congruence lattices of direct sums of algebrashttps://zbmath.org/1491.080032022-09-13T20:28:31.338867Z"Ghumashyan, Heghine"https://zbmath.org/authors/?q=ai:ghumashyan.heghine"Guričan, Jaroslav"https://zbmath.org/authors/?q=ai:gurican.jaroslavThe authors prove that in a congruence distributive variety the meet infinite distributivity of a congruence lattice is preserved by direct sums of algebras. It means that the congruence lattice of a direct sum of algebras is meet infinitely distributive if and only if a congruence lattice of every constituent algebra is meet infinitely distributive.
Reviewer: Ivan Chajda (Přerov)Lattices, spectral spaces, and closure operations on idempotent semiringshttps://zbmath.org/1491.160462022-09-13T20:28:31.338867Z"Jun, Jaiung"https://zbmath.org/authors/?q=ai:jun.jaiung"Ray, Samarpita"https://zbmath.org/authors/?q=ai:ray.samarpita"Tolliver, Jeffrey"https://zbmath.org/authors/?q=ai:tolliver.jeffreySummary: Spectral spaces, introduced by Hochster, are topological spaces homeomorphic to the prime spectra of commutative rings. In this paper we study spectral spaces in perspective of idempotent semirings which are algebraic structures receiving a lot of attention due to its several applications to tropical geometry. We first prove that a space is spectral if and only if it is the \textit{prime \(k\)-spectrum} of an idempotent semiring. In fact, we enrich Hochster's theorem by constructing a subcategory of idempotent semirings which is antiequivalent to the category of spectral spaces. We further provide examples of spectral spaces arising from sets of congruence relations of semirings. In particular, we prove that the \textit{space of valuations} and the \textit{space of prime congruences} on an idempotent semiring are spectral, and there is a natural bijection of sets between the two; this shows a stark difference between rings and idempotent semirings. We then develop several aspects of commutative algebra of semirings. We mainly focus on the notion of \textit{closure operations} for semirings, and provide several examples. In particular, we introduce an \textit{integral closure operation} and a \textit{Frobenius closure operation} for idempotent semirings.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.jorgConnectivity, 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.banaSome 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.Korovkin-type approximation theory in Riesz spaceshttps://zbmath.org/1491.460052022-09-13T20:28:31.338867Z"Dorai, Abderraouf"https://zbmath.org/authors/?q=ai:dorai.abderraouf"Chil, Elmiloud"https://zbmath.org/authors/?q=ai:chil.elmiloud"Wójtowicz, Marek"https://zbmath.org/authors/?q=ai:wojtowicz.marekSummary: In this paper, we prove the following Riesz spaces' version of the Korovkin theorem. Let \(E\) and \(F\) be two Archimedean Riesz spaces with \(F\) uniformly complete, let \(W\) be a nonempty subset of \(E^{+}\), and let \((T_{n})\) be a given sequence of \textbf{(r-u)-continuous} elements of \(\mathcal {L}(E,F)\), such that \(| T_{n}-T_{m}| x=| (T_{n}-T_{m})x|\) for all \(x\in E^{+},m,n\geq n_{0}\) (for a given \(n_{0}\in \mathbb {N} )\). If the sequence \((T_{n}x)_{n}(r-u)\)-converges for every \(x\in W\), then \((T_{n})(r-u)\)-converges also pointwise on the ideal \(E_{W}\), generated by \(W\), to a linear operator \(S_{0}:E_{W}\to F\). We also prove a similar Korovkin-type theorem for nets of operators. Some applications for \(f\)-algebras and orthomorphisms are presented.On the continuity of lattice isomorphisms on \(C(X,I)\)https://zbmath.org/1491.460182022-09-13T20:28:31.338867Z"Ehsani, Vahid"https://zbmath.org/authors/?q=ai:ehsani.vahid"Sady, Fereshteh"https://zbmath.org/authors/?q=ai:sady.fereshtehSummary: We investigate topological conditions on a compact Hausdorff space \(Y\), such that any lattice isomorphism \(\varphi:C(X, I)\rightarrow C(Y,I)\), where \(X\) is a compact Hausdorff space and \(I\) is the unit interval \([0,1]\), is continuous. It is shown that in either of cases that the set of \(G_\delta\) points of \(Y\) has a dense pseudocompact subset or \(Y\) does not contain the Stone-Čech compactification of \(\mathbb{N}\), such a lattice isomorphism is a homeomorphism.The fixed point property for closed neighborhoods of line segments in \(L^p\)https://zbmath.org/1491.470492022-09-13T20:28:31.338867Z"Schröder, Bernd S. W."https://zbmath.org/authors/?q=ai:schroder.bernd-s-wSummary: We prove that, in \(L^p\)-spaces with \(p \in (1, \infty]\), closed neighborhoods of line segments are dismantlable and hence every monotone operator on these neighborhoods has a fixed point. We also give an example that, for \(p = 1\), closed neighborhoods of line segments need not be dismantlable. It is an open question whether every monotone self map of a closed neighborhood of a line segment in \(L^1\) has a fixed point.Compactness and stability in diframeshttps://zbmath.org/1491.540302022-09-13T20:28:31.338867Z"Korkmaz, Esra"https://zbmath.org/authors/?q=ai:korkmaz.esra"Ertürk, Rıza"https://zbmath.org/authors/?q=ai:erturk.rizaSummary: The concept of diframe was introduced as a generalization of ditopological texture spaces. The purpose of this paper is to present the results of a study on the concepts of compactness and stability in the setting of diframes. Further, the bitopological concepts of locally compactness and locally stability are extended to diframes.Siegmund duality for continuous time Markov chains on \(\mathbb{Z}_+^d\)https://zbmath.org/1491.601332022-09-13T20:28:31.338867Z"Zhao, Pan"https://zbmath.org/authors/?q=ai:zhao.panSummary: For the continuous time Markov chain with transition function \(P(t)\) on \(\mathbb{Z}_+^d\), we give the necessary and sufficient conditions for the existence of its Siegmund dual with transition function \(\tilde{P}(t)\). If \(Q\), the \(q\)-matrix of \(P(t)\), is uniformly bounded, we show that the Siegmund dual relation can be expressed directly in terms of \(q\)-matrices, and a sufficient condition under which the \(Q\)-function is the Siegmund dual of some \(Q\)-function is also given.The log-approximate-rank conjecture is falsehttps://zbmath.org/1491.680732022-09-13T20:28:31.338867Z"Chattopadhyay, Arkadev"https://zbmath.org/authors/?q=ai:chattopadhyay.arkadev"Mande, Nikhil S."https://zbmath.org/authors/?q=ai:mande.nikhil-s"Sherif, Suhail"https://zbmath.org/authors/?q=ai:sherif.suhailAn improved dictatorship test with perfect completenesshttps://zbmath.org/1491.680832022-09-13T20:28:31.338867Z"Bhangale, Amey"https://zbmath.org/authors/?q=ai:bhangale.amey"Khot, Subhash"https://zbmath.org/authors/?q=ai:khot.subhash-a"Thiruvenkatachari, Devanathan"https://zbmath.org/authors/?q=ai:thiruvenkatachari.devanathanSummary: A Boolean function \(f:\{0,1\}^n\to\{0,1\}\) is called a dictator if it depends on exactly one variable i.e. \(f(x_1,x_2,\dots,x_n)=x_i\) for some \(i\in[n]\). In this work, we study a \(k\)-query dictatorship test. Dictatorship tests are central in proving many hardness results for constraint satisfaction problems. \par The dictatorship test is said to have perfect completeness if it accepts any dictator function. The soundness of a test is the maximum probability with which it accepts any function far from a dictator. Our main result is a \(k\)-query dictatorship test with perfect completeness and soundness \(\frac{2k+1}{2^k}\), where \(k\) is of the form \(2^t-1\) for any integer \(t>2\). This improves upon the result of [\textit{S. Tamaki} and \textit{Y. Yoshida}, Random Struct. Algorithms 47, No. 2, 386--406 (2015; Zbl 1341.68066)] which gave a dictatorship test with soundness \(\frac{2k+3}{2^k}\).
For the entire collection see [Zbl 1388.68010].Finding the subsets of variables of a partial Boolean function which are sufficient for its implementation in the classes defined by predicateshttps://zbmath.org/1491.680902022-09-13T20:28:31.338867Z"Parvatov, Nikolaĭ Georgievich"https://zbmath.org/authors/?q=ai:parvatov.nikolai-georgievichSummary: Given a class \(K\) of partial Boolean functions and a partial Boolean function \(f\) of \(n\) variables, a subset \(U\) of its variables is called \textit{sufficient for the implementation of} \(f\) in \(K\) if there exists an extension of \(f\) in \(K\) with arguments in \(U\). We consider the problem of recognizing all subsets sufficient for the implementation of \(f\) in \(K\). For some classes defined by relations, we propose the algorithms of solving this problem with complexity of \(O(2^nn^2)\) bit operations. In particular, we present some algorithms of this complexity for the class \(P_2^*\) of all partial Boolean functions and the class \(M_2^*\) of all monotone partial Boolean functions. The proposed algorithms use the Walsh-Hadamard and Möbius transforms.Logic and algebra in unfolded Petri nets: on a duality between concurrency and causal dependencehttps://zbmath.org/1491.681162022-09-13T20:28:31.338867Z"Bernardinello, Luca"https://zbmath.org/authors/?q=ai:bernardinello.luca"Ferigato, Carlo"https://zbmath.org/authors/?q=ai:ferigato.carlo"Pomello, Lucia"https://zbmath.org/authors/?q=ai:pomello.luciaA partially ordered set is a couple \((P, \leqslant )\), where \(\leqslant\) is an antisymetric, transitive and reflexive relation on \(P\). If \(x \leqslant y\) or \(y \leqslant x\) we say that \(x\) and \(y\) are comparable (a causal dependence relation); otherwise they are called concurrent (denoted by \( x\, \mathbf{co}\, y\), a concurrency relation). In this way two distinct orthogonality spaces are consequently obtained. \((P, \leqslant )\) is \textit{N-dense} iff for any \( x,y,z,t \in P\) such that \(x < y\), \(x <t\), \(z<t\), \( x\, \mathbf{co}\, z\), \(y\, \mathbf{co}\, z\) and \(y\, \mathbf{co}\, t\), there exists \(k \in P\) such that \(k\, \mathbf{co}\, y\), \(k\, \mathbf{co}\, z\), \( x < k\) and \(k < t\). When the condition of N-density holds on both these orthogonality spaces, orthomodular posets formed by closed sets, defined according to Dacey, are studied. It is shown that the condition originally imposed by Dacey on the orthogonality spaces for obtaining an orthomodular poset from his closed sets is in fact equivalent to N-density. The requirement of N-density was as well fundamental in a previous work on orthogonality spaces with the concurrency relation [\textit{L. Bernardinello} et al., Fundam. Inform. 105, No. 3, 211--235 (2010; Zbl 1209.68333)]. Starting from a partially ordered set modelling a concurrent process, dual results for orthogonality spaces with the causal dependence relation with respect to orthogonality spaces with the concurrency relation are obtained.
Reviewer: Damas Gruska (Bratislava)Hardness of rainbow coloring hypergraphshttps://zbmath.org/1491.681442022-09-13T20:28:31.338867Z"Guruswami, Venkatesan"https://zbmath.org/authors/?q=ai:guruswami.venkatesan"Saket, Rishi"https://zbmath.org/authors/?q=ai:saket.rishiSummary: A hypergraph is \(k\)-rainbow colorable if there exists a vertex coloring using \(k\) colors such that each hyperedge has all the \(k\) colors. Unlike usual hypergraph coloring, rainbow coloring becomes harder as the number of colors increases. This work studies the rainbow colorability of hypergraphs which are guaranteed to be nearly balanced rainbow colorable. Specifically, we show that for any \(Q\), \(k\ge 2\) and \(\ell\le k/2\), given a \(Qk\)-uniform hypergraph which admits a \(k\)-rainbow coloring satisfying: \par in each hyperedge \(e\), for some \(\ell_e\le\ell\) all but \(2\ell_e\) colors occur exactly \(Q\) times and the rest \((Q\pm-1)\) times, \par it is NP-hard to compute an independent set of \((1-\frac{\ell+1}{k}+\varepsilon)\)-fraction of vertices, for any constant \(\varepsilon>0\). In particular, this implies the hardness of even \((k/\ell)\)-rainbow coloring such hypergraphs. \par The result is based on a novel long code PCP test that ensures the strong balancedness property desired of the \(k\)-rainbow coloring in the completeness case. The soundness analysis relies on a mixing bound based on uniform reverse hypercontractivity due to Mossel, Oleszkiewicz, and Sen, which was also used in earlier proofs of the hardness of \(\omega(1)\)-coloring 2-colorable 4-uniform hypergraphs due to Saket, and \(k\)-rainbow colorable \(2k\)-uniform hypergraphs due to Guruswami and Lee.
For the entire collection see [Zbl 1388.68010].