Recent zbMATH articles in MSC 06Fhttps://zbmath.org/atom/cc/06F2023-12-07T16:00:11.105023ZWerkzeugOrders on magmas and computability theoryhttps://zbmath.org/1522.031722023-12-07T16:00:11.105023Z"Ha, Trang"https://zbmath.org/authors/?q=ai:ha.trang"Harizanov, Valentina"https://zbmath.org/authors/?q=ai:harizanov.valentina-sSummary: We investigate algebraic and computability-theoretic properties of orderable magmas. A magma is an algebraic structure with a single binary operation. A right order on a magma is a linear ordering of its domain, which is right-invariant with respect to the magma operation. We use tools of computability theory to investigate Turing complexity of orders on computable orderable magmas. A magma is computable if it is finite, or if its domain can be identified with the set of natural numbers and the magma operation is computable. Interesting orderable magmas that are not even associative come from knot theory.Characterization of idempotent \(n\)-uninormshttps://zbmath.org/1522.032712023-12-07T16:00:11.105023Z"Mesiarová-Zemánková, Andrea"https://zbmath.org/authors/?q=ai:mesiarova-zemankova.andreaSummary: The structure of idempotent \(n\)-uninorms is studied. We show that each idempotent 2-uninorm can be expressed as an ordinal sum of an idempotent uninorm (possibly also of a countable number of idempotent semigroups with operations min and max) and a 2-uninorm from Class 1 (possibly restricted to open or half-open unit square). Similar results are shown also for idempotent \(n\)-uninorms. Further, it is shown that idempotent \(n\)-uninorms are in one-to-one correspondence with special lower semi-lattices defined on the unit interval. The \(z\)-ordinal sum construction for partially ordered semigroups is also defined.Valued quantum B-algebrashttps://zbmath.org/1522.033332023-12-07T16:00:11.105023Z"Ciungu, Lavinia Corina"https://zbmath.org/authors/?q=ai:ciungu.lavinia-corinaSummary: The aim of this paper is to define the notion of pseudo-valuations on quantum B-algebras and to investigate their properties. We also give characterizations of pseudo-valuations and we show that there is a relationship between the positive strong pseudo-valuations and the filters of a unital quantum B-algebra. We define and characterize the pseudo-valuations on the particular cases of quantum B-algebras with product and pointed quantum B-algebras. We also prove extension theorems for pseudo-valuations on good quantum B-algebras satisfying Glivenko property, and we investigate certain possible generalizations of Horn-Tarski extension theorem for the case of pseudo-valuations on quantales.Representation of perfect and \(n\)-perfect pseudo effect algebrashttps://zbmath.org/1522.033342023-12-07T16:00:11.105023Z"Dvurečenskij, Anatolij"https://zbmath.org/authors/?q=ai:dvurecenskij.anatolij|dvurechenskii.anatolii-vasilevichSummary: A perfect (an \(n\)-perfect) pseudo effect algebra can be decomposed into two \((n+1\) many) non-empty and mutually comparable slices. They generalize perfect MV-algebras studied in [\textit{A. Di Nola} and \textit{A. Lettieri}, Stud. Log. 53, No. 3, 417--432 (1994; Zbl 0812.06010)]. We characterize such a pseudo effect algebra as an interval in the semidirect product of the po-group \(\mathbb{Z}\) or \(\frac{1}{n} \mathbb{Z}\) with a directed po-group \(G\) satisfying a stronger type of the Riesz Decomposition Property, \(RDP_1\), and the semidirect product is ordered lexicographically. We show that the category of perfect and the category of \(n\)-perfect pseudo effect algebras with \(RDP_1\) are categorically equivalent to a special category of directed po-groups satisfying \(RDP_1\).On generalized subreducts of Tarski's algebras of relations with the operation of bi-directional intersectionhttps://zbmath.org/1522.033392023-12-07T16:00:11.105023Z"Bredikhin, Dmitry A."https://zbmath.org/authors/?q=ai:bredikhin.dmitrii-aleksandrovichSummary: In the paper, the systems of axioms for classes of groupoids and partially ordered groupoids of relations with the operation of bi-directional intersection are found.On bounded residuated \(\ell\)EQ-algebrashttps://zbmath.org/1522.033472023-12-07T16:00:11.105023Z"Luan, Wei"https://zbmath.org/authors/?q=ai:luan.wei"Yang, Yichuan"https://zbmath.org/authors/?q=ai:yang.yichuanSummary: An EQ-algebra has three basic binary operations (meet, multiplication and a fuzzy equality) and a top element. An \(\ell\)EQ-algebra is a lattice-ordered EQ-algebra satisfying the substitution property of the join operation. In this article, we study bounded residuated \(\mathcal{l}\) EQ-algebras (BR-\(\ell\)EQ-algebras for short). We introduce a subvariety RL-EQ-algebras of BR-\(\ell\)EQ-algebras, and prove that the categories of RL-EQ-algebras and residuated lattices are categorical isomorphic. We also prove that RL-EQ-algebras are precisely the BR-\(\ell\)EQ-algebras that can be reconstructed from residuated lattices. We further show the existence of a closure operator on the poset of all BR-\(\ell\)EQ-algebras with the same lattice and multiplication reduct, the existence of the maximum element in the poset. Then we introduce filters in BR-\(\ell\)EQ-algebras and give a lattice isomorphism between the filter lattice and the congruence lattice. Finally, we prove that the category of residuated lattices is isomorphic to a reflective subcategory of BR-\(\ell\)EQ-algebras.The extension of L-algebras and stateshttps://zbmath.org/1522.033482023-12-07T16:00:11.105023Z"Mao, Lingling"https://zbmath.org/authors/?q=ai:mao.lingling"Xin, Xiaolong"https://zbmath.org/authors/?q=ai:xin.xiaolong"Zhang, Shunli"https://zbmath.org/authors/?q=ai:zhang.shunliSummary: In this paper, we introduce EL-algebra, a new algebraic structure extended from L-algebras, where the top element is not assumed and each EL-algebra contains an L-algebra. First, we study the dual atom and the branch on EL-algebras, from which we find that the implication of any two elements in an EL-algebra is always in branch \(V(1)\), which is an L-algebra. Based on this result, some relationships among EL-algebras, L-algebras, BCK-algebras, and \(l\)-groups are presented. Moreover, we introduce and investigate prime EL-algebras, ideals and congruence relations on EL-algebras. Then, the notions of Bosbach state and state-morphism on EL-algebras are presented, and some of their properties are investigated. Finally, using two different kinds of orthogonal relations on EL-algebras, \(\top\) and \(\bot\), we define two classes of Riečan states on EL-algebras. In addition, some relationships between the Bosbach state and these two Riečan states are studied.The Archimedean property: new horizons and perspectiveshttps://zbmath.org/1522.060222023-12-07T16:00:11.105023Z"Ledda, Antonio"https://zbmath.org/authors/?q=ai:ledda.antonio"Paoli, Francesco"https://zbmath.org/authors/?q=ai:paoli.francesco"Tsinakis, Constantine"https://zbmath.org/authors/?q=ai:tsinakis.constantineSummary: Although there have been repeated attempts to define the concept of an Archimedean algebra for individual classes of residuated lattices, there is no all-purpose definition that suits the general case. We suggest as a possible candidate the notion of a normal-valued and \(e\)-cyclic residuated lattice that has the \textit{zero radical compact property} -- namely, a normal-valued and \(e\)-cyclic residuated lattice in which every principal convex subuniverse has a trivial radical (understood as the intersection of all its maximal convex subuniverses). We characterize the Archimedean members in the variety of \(e\)-cyclic residuated lattices, as well as in various special cases of interest. A theorem to the effect that each Archimedean and prelinear GBL-algebra is commutative, subsuming as corollaries several analogous results from the recent literature, is grist to the mill of our proposal's adequacy. Finally, we revisit the concept of a \textit{hyper-Archimedean} residuated lattice, another notion with which researchers have engaged from disparate angles, and investigate some of its properties.On ordinal sums of partially ordered monoids: a unified approach to ordinal sum constructions of t-norms, t-conorms and uninormshttps://zbmath.org/1522.060262023-12-07T16:00:11.105023Z"Dvořák, Antonín"https://zbmath.org/authors/?q=ai:dvorak.antonin"Holčapek, Michal"https://zbmath.org/authors/?q=ai:holcapek.michal"Paseka, Jan"https://zbmath.org/authors/?q=ai:paseka.janSummary: This paper introduces two fundamental types of ordinal sum constructions for partially ordered monoids that are determined by two specific partial orderings on the disjoint union of the partially ordered monoids. Both ordinal sums of partially ordered monoids are generalized with the help of operators on posets, which combine, in some sense, the properties of interior and closure operators on posets. The proposed approach provides a unified view on several known constructions of ordinal sums of t-norms and t-conorms on posets (lattices) and introduces generalized ordinal sums of uninorms on posets (lattices).Correction to: ``Group representation for even and odd involutive commutative residuated chains''https://zbmath.org/1522.060272023-12-07T16:00:11.105023Z"Jenei, Sándor"https://zbmath.org/authors/?q=ai:jenei.sandorCorrection to the author's paper [ibid. 110, No. 4, 881--922 (2022; Zbl 1509.06008)].Reticulation of a quantale, pure elements and new transfer propertieshttps://zbmath.org/1522.060282023-12-07T16:00:11.105023Z"Georgescu, George"https://zbmath.org/authors/?q=ai:georgescu.georgeSummary: We know from a previous paper that the reticulation of a coherent quantale \(A\) is a bounded distributive lattice \(L(A)\) whose prime spectrum is homeomorphic to \(m\)-prime spectrum of \(A\). This paper studies how the reticulation can be used for transferring some properties of bounded distributive lattices to quantales and vice versa. We shall illustrate this thesis by proving several results on the pure and \(w\)-pure elements of the quantale \(A\) by means of the reticulation \(L(A)\). In particular, we shall investigate how the properties of \(\sigma\)-ideals of \(L(A)\) can be transferred to pure and \(w\)-pure elements of \(A\). Then the obtained transfer properties are used to prove new algebraic and topological results and characterization theorems for some important classes of quantales: normal quantales, \(mp\)-quantales, \(PF\)-quantales, purified quantales and \(PP\)-quantales.MIX \(\star\)-autonomous quantales and the continuous weak orderhttps://zbmath.org/1522.060292023-12-07T16:00:11.105023Z"Gouveia, Maria João"https://zbmath.org/authors/?q=ai:gouveia.maria-joao"Santocanale, Luigi"https://zbmath.org/authors/?q=ai:santocanale.luigiSummary: The set of permutations on a finite set can be given a lattice structure (known as the weak Bruhat order). The lattice structure is generalized to the set of words on a fixed alphabet \(\varSigma=\{\,x,y,z,\ldots\,\}\), where each letter has a fixed number of occurrences (these lattices are known as multinomial lattices and, in dimension 2, as lattices of lattice paths). By interpreting the letters \(x,y,z,\ldots\) as axes, these words can be interpreted as discrete increasing paths on a grid of a \(d\)-dimensional cube, where \(d=\text{card}(\varSigma)\).
We show in this paper how to extend this order to images of continuous monotone paths from the unit interval to a \(d\)-dimensional cube. The key tool used to realize this construction is the quantale \(\mathsf{L}_{\vee}(\mathbb{I})\) of join-continuous functions from the unit interval to itself; the construction relies on a few algebraic properties of this quantale: it is \(\star\)-autonomous and it satisfies the mix rule.
We begin developing a structural theory of these lattices by characterizing join-irreducible elements, and by proving these lattices are generated from their join-irreducible elements under infinite joins.
For the entire collection see [Zbl 1398.68023].Constructing \(Q\)-algebras from \(Q\)-moduleshttps://zbmath.org/1522.060302023-12-07T16:00:11.105023Z"Han, Shengwei"https://zbmath.org/authors/?q=ai:han.shengweiSummary: The main purpose of this paper is to consider the construction of \(Q\)-algebras from \(Q\)-modules. First, we investigate the relationship between bimodules over a quantale \(Q\) and double quantale modules over \(Q\). Then, we give different ways of constructing \(Q\)-algebras from \(Q\)-modules, which generalize the work of Rosenthal for constructing quantales (see Lemma 3.18 and Corollary 3.20).Reflectors to quantaleshttps://zbmath.org/1522.060312023-12-07T16:00:11.105023Z"Zhang, Xia"https://zbmath.org/authors/?q=ai:zhang.xia"Paseka, Jan"https://zbmath.org/authors/?q=ai:paseka.jan"Feng, Jianjun"https://zbmath.org/authors/?q=ai:feng.jianjun"Chen, Yudong"https://zbmath.org/authors/?q=ai:chen.yudongSummary: In this paper, we show that marked quantales have a reflection into quantales. To obtain the reflection we construct free quantales over marked quantales using appropriate lower sets. A marked quantale is a posemigroup in which certain admissible subsets are required to have joins, and multiplication distributes over these. Sometimes are the admissible subsets in question specified by means of a so-called selection function. A distinguishing feature of the study of marked quantales is that a small collection of axioms of an elementary nature allows one to do much that is traditional at the level of quantales. The axioms are sufficiently general to include as examples of marked quantales the classes of posemigroups, \(\sigma\)-quantales, prequantales and quantales. Furthermore, we discuss another reflection to quantales obtained by the injective hull of a posemigroup.o-groups of finite rank and divisibility in their groups of o-automorphismshttps://zbmath.org/1522.060322023-12-07T16:00:11.105023Z"Lafuente-Rodriguez, Ramiro H."https://zbmath.org/authors/?q=ai:lafuente-rodriguez.ramiro-hSummary: We provide a full description of totally ordered groups of finite archimedean rank and study solvability of equations of the form \(x^n=g\) in the group of ordered automorphisms of an a-closed totally ordered group \(G\) of finite archimedean rank. We also give a full description of these groups of o-automorphisms and a characterization of the elements of a particular group that have an \(n\)-th root.Pseudo L-algebrashttps://zbmath.org/1522.060332023-12-07T16:00:11.105023Z"Xin, X. L."https://zbmath.org/authors/?q=ai:xin.xiaolong"Yang, X. F."https://zbmath.org/authors/?q=ai:yang.xiaofei"Ma, Y. C."https://zbmath.org/authors/?q=ai:ma.yingcang(no abstract)Fuzziness in \(L\)-algebrashttps://zbmath.org/1522.060342023-12-07T16:00:11.105023Z"Zhan, Qiuyan"https://zbmath.org/authors/?q=ai:zhan.qiuyan"Wang, Jing"https://zbmath.org/authors/?q=ai:wang.jing.172"Luan, Wei"https://zbmath.org/authors/?q=ai:luan.wei"Yang, Yichuan"https://zbmath.org/authors/?q=ai:yang.yichuanSummary: The aim of this paper is to introduce fuzzy ideals and fuzzy (weak) congruence relations of \(L\)-algebras. We give a sufficient and necessary condition for a fuzzy set to be a fuzzy ideal of an \(L\)-algebra. After obtaining a necessary condition for a fuzzy set to be a fuzzy ideal of \(L\)-algebras, we construct an example to show that the condition is not sufficient. For \textit{CKL}-algebras, however, we prove that it is both necessary and sufficient. Then we give a process to generate a fuzzy ideal by a fuzzy set of \textit{CKL}-algebras. Finally, we establish a correspondence between fuzzy ideals and fuzzy weak congruence relations of \(L\)-algebras, and get homomorphism theorems induced by fuzzy congruence relations.Monoids with quantale-valued preorders: globalizations and localizationshttps://zbmath.org/1522.180092023-12-07T16:00:11.105023Z"Zhao, Xiaojuan"https://zbmath.org/authors/?q=ai:zhao.xiaojuanSummary: We consider compatible monoid structures on a \(\mathsf{DQ}\)-category, where \(\mathsf{Q}\) is a commutative and divisible quantale, and \(\mathsf{DQ}\) is the quantaloid of diagonals of \(\mathsf{Q}\). Such structures, called \(\mathsf{DQ}\)-monoids, may be treated as monoids equipped with a preorder valued in \(\mathsf{Q}\), whose elements are not supposed to exist globally. The globalization functors map \(\mathsf{DQ}\)-monoids to global ones, i.e., monoids on a \(\mathsf{Q}\)-category, called \(\mathsf{Q}\)-monoids. Conversely, \(\mathsf{Q}\)-monoids over \(\mathsf{Q}\) give rise to \(\mathsf{DQ}\)-monoids via the localization functors. The interactions between the globalization functors and the localization functors are investigated. In particular, a necessary and sufficient condition on \(\mathsf{Q}\) is provided such that the localizations are reversible by the globalizations.On presheaf submonads of quantale-enriched categorieshttps://zbmath.org/1522.180122023-12-07T16:00:11.105023Z"Clementino, Maria Manuel"https://zbmath.org/authors/?q=ai:clementino.maria-manuel"Fitas, Carlos"https://zbmath.org/authors/?q=ai:fitas.carlosThe authors' abstract says: ``This paper focuses on the presheaf monad, or the free cocompletion monad, and its submonads on the realm of \(V\)-categories, for a quantale \(V\). First we present two characterisations of presheaf submonads, both using \(V\)-distributors: one based on admissible classes of \(V\)-distributors, and other using Beck-Chevalley conditions on \(V\)-distributors. Further we prove that lax idempotency for 2-monads on \(V\)-Cat can be characterized via such a Beck-Chevalley condition. Then we focus on the study of the Eilenberg-Moore categories of algebras for our monads, having as main examples the formal ball monad and the Lawvere-Cauchy completion monad.''
Here \(V\) is a complete lattice equipped with a commutative monoid structure such that the monoid operation, written as \(\otimes\), is distributive with respect to arbitrary joins; following the quantale theory terminology, the authors also say that \(V\) is a commutative unital quantale, and briefly call it just a quantale. The paper substantially contributes to the theory of quantale enriched categories, whose examples include the category of (pre)ordered sets and variations of the category of metric spaces among many others.
Reviewer: George Janelidze (Cape Town)Generic algebraic properties in spaces of enumerated groupshttps://zbmath.org/1522.201272023-12-07T16:00:11.105023Z"Goldbring, Isaac"https://zbmath.org/authors/?q=ai:goldbring.isaac"Kunnawalkam Elayavalli, Srivatsav"https://zbmath.org/authors/?q=ai:kunnawalkam-elayavalli.srivatsav"Lodha, Yash"https://zbmath.org/authors/?q=ai:lodha.yashThe paper under review is a contribute to the study of countable groups from a topological point of view. An enumerated group is a group structure on the set \(\mathbb{N}\) of natural numbers. Equipped with a natural topology (defined in Section 3), the set of all enumerated groups forms a Polish space which is very natural from the point of view of first-order logic in that it is simply the space of countably infinite \(\mathcal{L}\)-structures in the case that \(\mathcal{L}\) is the usual first-order language of groups.
The Baire category theorem states that, in any Polish space \(X\), the intersection of countably many dense open subsets of \(X\) is once again a dense subset of \(X\). An intersection of countably many dense open sets is called comeager, and if a certain property holds for all elements of a comeager subset of the space, it is natural to say that the property is generic.
The authors, ``using elementary tools and well-known examples from combinatorial group theory (combined with the Baire category theorem), obtain a plethora of results demonstrating that several phenomena in group theory are generic''. They also provide ``a connection between genericity in these spaces, the word problem for finitely generated groups and model-theoretic forcing. Using these connections, they investigate a natural question raised by \textit{D. Osin} [Ann. Math. (2) 172, No. 1, 1--39 (2010; Zbl 1203.20031)]: when does a certain space of enumerated groups contain a comeager isomorphism class?'' The authors obtain a sufficient condition that allows us to answer Osin's question in the negative for the space of all enumerated groups and the space of left orderable enumerated groups.
Reviewer: Egle Bettio (Venezia)Semiring identities of finite inverse semigroupshttps://zbmath.org/1522.202292023-12-07T16:00:11.105023Z"Gusev, Sergey V."https://zbmath.org/authors/?q=ai:gusev.sergei-valentinovich"Volkov, Mikhail V."https://zbmath.org/authors/?q=ai:volkov.mikhail-vI cite the authors' abstract, which contains the important information about this (naturally quite technical) paper: ``We study the Finite Basis Problem for finite additively idempotent (ai) semirings whose multiplicative reducts are inverse semigroups. In particular, we show that each additively idempotent semiring whose multiplicative reduct is a nontrivial rook monoid admits no finite identity basis, and so do almost all additively idempotent semirings whose multiplicative reducts are combinatorial inverse semigroups.''
The paper answers among others the following question: Which finite naturally semilattice-ordered inverse semigroups are finitely based, in either of the signatures \(\{+,\cdot\}\) or \(\{+, \cdot, 0\}\)?
Reviewer: Ulrich Knauer (Oldenburg)Generalized torsion for hyperbolic 3-manifold groups with arbitrary large rankhttps://zbmath.org/1522.570132023-12-07T16:00:11.105023Z"Ito, Tetsuya"https://zbmath.org/authors/?q=ai:ito.tetsuya"Motegi, Kimihiko"https://zbmath.org/authors/?q=ai:motegi.kimihiko"Teragaito, Masakazu"https://zbmath.org/authors/?q=ai:teragaito.masakazuSummary: Let \(G\) be a group and \(g\) a non-trivial element in \(G\). If some non-empty finite product of conjugates of \(g\) equals to the trivial element, then \(g\) is called a \textit{generalized torsion element}. To the best of our knowledge, we have no hyperbolic 3-manifold groups with generalized torsion elements whose rank is explicitly known to be greater than two. The aim of this short note is to demonstrate that for a given integer \(n > 1\) there are infinitely many closed hyperbolic 3-manifolds \(M_n\) which enjoy the property: (i) the Heegaard genus of \(M_n\) is \(n\), (ii) the rank of \(\pi_1(M_n)\) is \(n\), and (ii) \(\pi_1(M_n)\) has a generalized torsion element. Furthermore, we may choose \(M_n\) as homology lens spaces and so that the order of the generalized torsion element is arbitrarily large.
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