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10 pages, 313 KiB

Open AccessArticle

Topologies on Smashed Twisted Wreath Products of Metagroups

by Sergey Victor Ludkowski

Axioms 2023, 12(3), 240; https://doi.org/10.3390/axioms12030240 - 25 Feb 2023

Cited by 1 | Viewed by 665

Abstract

In this article, topologies on metagroups and quasigroups are studied. Topologies on smashed twisted wreath products of metagroups are scrutinized, which are making them topological metagroups. For this purpose, transversal sets are studied. As a tool for this, semi-direct products of topological metagroups [...] Read more.

In this article, topologies on metagroups and quasigroups are studied. Topologies on smashed twisted wreath products of metagroups are scrutinized, which are making them topological metagroups. For this purpose, transversal sets are studied. As a tool for this, semi-direct products of topological metagroups are also investigated. They have specific features in comparison with topological groups because of the nonassociativity, in general, of metagroups. A related structure of topological metagroups is investigated. Particularly, their compact subloops and submetagroups are studied. Isomorphisms of topological unital quasigroups (i.e., loops) obtained by the smashed twisted wreath products are investigated. Examples are provided. Full article

(This article belongs to the Special Issue Algebra, Logic and Applications)

21 pages, 337 KiB

Open AccessArticle

Representations, Translations and Reductions for Ternary Semihypergroups

by Anak Nongmanee and Sorasak Leeratanavalee

Axioms 2022, 11(11), 626; https://doi.org/10.3390/axioms11110626 - 08 Nov 2022

Cited by 1 | Viewed by 1055

Abstract

The concept of ternary semihypergroups can be considered as a natural generalization of arbitrary ternary semigroups. In fact, each ternary semigroup can be constructed to a ternary semihypergroup. In this article, we investigate some interesting algebraic properties of ternary semihypergroups induced by semihypergroups. [...] Read more.

The concept of ternary semihypergroups can be considered as a natural generalization of arbitrary ternary semigroups. In fact, each ternary semigroup can be constructed to a ternary semihypergroup. In this article, we investigate some interesting algebraic properties of ternary semihypergroups induced by semihypergroups. Then, we extend the well-known result on group theory and semigroup theory, the so-called Cayley’s theorem, to study on ternary semihypergroups. This leads us to construct the ternary semihypergroups of all multivalued full binary functions. In particular, we investigate that each element of a ternary semihypergroup induced by a semihypergroup can be represented by a multivalued full binary function. Moreover, we introduce the concept of translations for ternary semihypergroups which can be considered as a generalization of translations on ternary semigrgoups. Then, we construct ternary semihypergroups of all multivalued full functions and ternary semihypergroups via translations. So, some interesting algebraic properties are investigated. At the last section, we discover that there are ternary semihypergroups satisfying some significant conditions which can be reduced to semihypergroups. Furthermore, ternary semihypergroups with another one condition can be reduced to idempotent semihypergroups. Full article

(This article belongs to the Special Issue Algebra, Logic and Applications)

13 pages, 330 KiB

Open AccessArticle

Some Implicational Semilinear Gaggle Logics: (Dual) Residuated-Connected Logics

by Eunsuk Yang

Axioms 2022, 11(4), 183; https://doi.org/10.3390/axioms11040183 - 18 Apr 2022

Viewed by 1411

Abstract

Implicational partial Galois logics and some of their semilinear extensions, such as semilinear extensions satisfying abstract Galois and dual Galois connection properties, have been introduced together with their relational semantics. However, similar extensions satisfying residuated, dual residuated connection properties have not. This paper [...] Read more.

Implicational partial Galois logics and some of their semilinear extensions, such as semilinear extensions satisfying abstract Galois and dual Galois connection properties, have been introduced together with their relational semantics. However, similar extensions satisfying residuated, dual residuated connection properties have not. This paper fills the gaps by introducing those semilinear extensions and their relational semantics. To this end, the class of implicational (dual) residuated-connected prelinear gaggle logics is defined and it is verified that these logics are semilinear. In particular, associated with the contribution of this work, we note the following two: One is that implications can be introduced by residuated connection in semilinear logics. This shows that the residuated, dual residuated connection properties are important and so need to be investigated in semilinear logics. The other is that set-theoretic relational semantics can be provided for semilinear logics. Semilinear logics have been dealt with extensively in algebraic context, whereas they have not yet been performed in the set-theoretic one. Full article

(This article belongs to the Special Issue Algebra, Logic and Applications)

20 pages, 366 KiB

Open AccessArticle

Two Open Problems on CA-Groupoids and Cancellativities of T2CA-Groupoids

by Xiaogang An, Xiaohong Zhang and Zhirou Ma

Axioms 2022, 11(4), 169; https://doi.org/10.3390/axioms11040169 - 08 Apr 2022

Cited by 3 | Viewed by 1834

Abstract

Cyclic associative groupoids (CA-groupoids) and Type-2 cyclic associative groupoids (T2CA-groupoids) are two types of non-associative groupoids which satisfy cyclic associative law and type-2 cyclic associative law, respectively. In this paper, we prove two theorems that weak cancellativity is cancellativity and right quasi-cancellativity is [...] Read more.

Cyclic associative groupoids (CA-groupoids) and Type-2 cyclic associative groupoids (T2CA-groupoids) are two types of non-associative groupoids which satisfy cyclic associative law and type-2 cyclic associative law, respectively. In this paper, we prove two theorems that weak cancellativity is cancellativity and right quasi-cancellativity is left quasi-cancellativity in a CA-groupoid, thus successfully solving two open problems. Moreover, the relationships among separativity, quasi-cancellativity and commutativity in a CA-groupoid are discussed. Finally, we study the various cancellativities of T2CA-groupoids such as power cancellativity, quasi-cancellativity and cancellativity. By determining the relationships between them, we can illuminate the structure of T2CA-groupoids. Full article

(This article belongs to the Special Issue Algebra, Logic and Applications)

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20 pages, 373 KiB

Open AccessFeature PaperArticle

Splitting Extensions of Nonassociative Algebras and Modules with Metagroup Relations

by Sergey Victor Ludkowski

Axioms 2022, 11(3), 131; https://doi.org/10.3390/axioms11030131 - 14 Mar 2022

Cited by 1 | Viewed by 1432

Abstract

A class of nonassociative algebras is investigated with mild relations induced from metagroup structures. Modules over nonassociative algebras are studied. For a class of modules over nonassociative algebras, their extensions and splitting extensions are scrutinized. For this purpose tensor products of modules and [...] Read more.

A class of nonassociative algebras is investigated with mild relations induced from metagroup structures. Modules over nonassociative algebras are studied. For a class of modules over nonassociative algebras, their extensions and splitting extensions are scrutinized. For this purpose tensor products of modules and induced modules over nonassociative algebras are investigated. Moreover, a developed cohomology theory on them is used. Full article

(This article belongs to the Special Issue Algebra, Logic and Applications)

10 pages, 230 KiB

Open AccessArticle

Forward Order Law for the Reflexive Inner Inverse of Multiple Matrix Products

by Wanna Zhou, Zhiping Xiong and Yingying Qin

Axioms 2022, 11(3), 123; https://doi.org/10.3390/axioms11030123 - 10 Mar 2022

Cited by 2 | Viewed by 1541

Abstract

The generalized inverse has numerous important applications in aspects of the theoretic research of matrices and statistics. One of the core problems of generalized inverse is finding the necessary and sufficient conditions for the reverse (or the forward) order laws for the generalized [...] Read more.

The generalized inverse has numerous important applications in aspects of the theoretic research of matrices and statistics. One of the core problems of generalized inverse is finding the necessary and sufficient conditions for the reverse (or the forward) order laws for the generalized inverse of matrix products. In this paper, by using the extremal ranks of the generalized Schur complement, some necessary and sufficient conditions are given for the forward order law for

A_{1} {1, 2} A_{2} {1, 2} \dots A_{n} {1, 2} \subseteq (A_{1} A_{2} \dots A_{n}) {1, 2}

. Full article

(This article belongs to the Special Issue Algebra, Logic and Applications)

21 pages, 343 KiB

Open AccessArticle

QM-BZ-Algebras and Quasi-Hyper BZ-Algebras

by Yudan Du and Xiaohong Zhang

Axioms 2022, 11(3), 93; https://doi.org/10.3390/axioms11030093 - 24 Feb 2022

Cited by 8 | Viewed by 1876

Abstract

B Z

-algebra, as the common generalization of

B C I

-algebra and

B C C

-algebra, is a kind of important logic algebra. Herein, the new concepts of QM-

B Z

-algebra and quasi-hyper

B Z

-algebra are proposed and their structures [...] Read more.

B Z

-algebra, as the common generalization of

B C I

-algebra and

B C C

-algebra, is a kind of important logic algebra. Herein, the new concepts of QM-

B Z

-algebra and quasi-hyper

B Z

-algebra are proposed and their structures and constructions are studied. First, the definition of QM-

B Z

-algebra is presented, and the structure of QM-

B Z

-algebra is obtained: Each QM-

B Z

-algebra is KG-union of quasi-alter

B C K

-algebra and anti-grouped

B Z

-algebra. Second, the new concepts of generalized quasi-left alter (hyper)

B Z

-algebras and QM-hyper

B Z

-algebra are introduced, and some characterizations of them are investigated. Third, the definition of quasi-hyper

B Z

-algebra is proposed, and the relationships among

B Z

-algebra, hyper

B Z

-algebra, quasi-hyper

B C I

-algebra, and quasi-hyper

B Z

-algebra are discussed. Finally, several special classes of quasi-hyper

B Z

-algebras are studied in depth and the following important results are proved: (1) an anti-grouped quasi-hyper

B Z

-algebra is an anti-grouped

B Z

-algebra; (2) every generalized anti-grouped quasi-hyper

B Z

-algebra corresponds to a semihypergroup. Full article

(This article belongs to the Special Issue Algebra, Logic and Applications)

18 pages, 365 KiB

Open AccessArticle

On Cohomology of Simple Modules for Modular Classical Lie Algebras

by Sherali S. Ibraev, Larissa S. Kainbaeva and Saulesh K. Menlikozhaeva

Axioms 2022, 11(2), 78; https://doi.org/10.3390/axioms11020078 - 16 Feb 2022

Cited by 1 | Viewed by 1978

Abstract

In this article, we obtain some cohomology of classical Lie algebras over an algebraically closed field of characteristic

p > h,

where

h

is a Coxeter number, with coefficients in simple modules. We assume that these classical Lie algebras are Lie algebras [...] Read more.

In this article, we obtain some cohomology of classical Lie algebras over an algebraically closed field of characteristic

p > h,

where

h

is a Coxeter number, with coefficients in simple modules. We assume that these classical Lie algebras are Lie algebras of semisimple and simply connected algebraic groups. To describe the cohomology of simple modules, we will use the properties of the connections between ordinary and restricted cohomology of restricted Lie algebras. Full article

(This article belongs to the Special Issue Algebra, Logic and Applications)

20 pages, 326 KiB

Open AccessArticle

A Class of BCI-Algebra and Quasi-Hyper BCI-Algebra

by Xiaohong Zhang and Yudan Du

Axioms 2022, 11(2), 72; https://doi.org/10.3390/axioms11020072 - 10 Feb 2022

Cited by 9 | Viewed by 2285

Abstract

In this paper, we study the connection between generalized quasi-left alter

B C I

-algebra and commutative Clifford semigroup by introducing the concept of an adjoint semigroup. We introduce QM-

B C I

algebra, in which every element is a quasi-minimal element, and [...] Read more.

In this paper, we study the connection between generalized quasi-left alter

B C I

-algebra and commutative Clifford semigroup by introducing the concept of an adjoint semigroup. We introduce QM-

B C I

algebra, in which every element is a quasi-minimal element, and prove that each QM-

B C I

algebra is equivalent to generalized quasi-left alter

B C I

-algebra. Then, we introduce the notion of generalized quasi-left alter-hyper

B C I

-algebra and prove that every generalized quasi-left alter-hyper

B C I

-algebra is a generalized quasi-left alter

B C I

-algebra. Next, we propose a new notion of quasi-hyper

B C I

algebra and discuss the relationship among them. Moreover, we study the subalgebras of quasi-hyper

B C I

algebra and the relationships between

H_{v}

-group and quasi-hyper

B C I

-algebra, hypergroup and quasi-hyper

B C I

-algebra. Finally, we propose the concept of a generalized quasi-left alter quasi-hyper

B C I

algebra and QM-quasi hyper

B C I

-algebra and discuss the relationships between them and related

B C I

-algebra. Full article

(This article belongs to the Special Issue Algebra, Logic and Applications)

11 pages, 4336 KiB

Open AccessFeature PaperArticle

Computational Experiments with the Roots of Fibonacci-like Polynomials as a Window to Mathematics Research

by Sergei Abramovich, Nikolay V. Kuznetsov and Gennady A. Leonov

Axioms 2022, 11(2), 48; https://doi.org/10.3390/axioms11020048 - 26 Jan 2022

Viewed by 3120

Abstract

Fibonacci-like polynomials, the roots of which are responsible for a cyclic behavior of orbits of a second-order two-parametric difference equation, are considered. Using Maple and Wolfram Alpha, the location of the largest and the smallest roots responsible for the cycles of period [...] Read more.

Fibonacci-like polynomials, the roots of which are responsible for a cyclic behavior of orbits of a second-order two-parametric difference equation, are considered. Using Maple and Wolfram Alpha, the location of the largest and the smallest roots responsible for the cycles of period p among the roots responsible for the cycles of periods 2^kp (period-doubling) and kp (period-multiplying) has been determined. These purely computational results of experimental mathematics, made possible by the use of modern digital tools, can be used as a motivation for confirmation through not-yet-developed methods of formal mathematics. Full article

(This article belongs to the Special Issue Algebra, Logic and Applications)

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Review

Jump to: Research

11 pages, 270 KiB

Open AccessReview

Logic, Game Theory, and Social Choice: What Do They Have in Common?

by Harrie de Swart

Axioms 2022, 11(10), 518; https://doi.org/10.3390/axioms11100518 - 30 Sep 2022

Viewed by 1157

Abstract

The answer to the question above is that in all these domains axiomatic characterizations are given of, respectively, mathematical reasoning, certain notions from game theory, and certain social choice rules. The meaning of the completeness theorem in logic is that mathematical reasoning can [...] Read more.

The answer to the question above is that in all these domains axiomatic characterizations are given of, respectively, mathematical reasoning, certain notions from game theory, and certain social choice rules. The meaning of the completeness theorem in logic is that mathematical reasoning can be characterized by a handful of certain (logical) axioms and rules. If we apply mathematical reasoning to elementary arithmetic, i.e., the addition and multiplication of natural numbers, it turns out that almost all true arithmetical statements, for instance,

\forall x \forall y [x + y = y + x]

, can be logically deduced from the axioms of Peano. However, in 1931 Kurt Gődel showed that the axioms of Peano do not (fully) characterize the addition and multiplication of the natural numbers, more precisely, that there are certain special self-referential arithmetical sentences that, although true, cannot be deduced from Peano’s axioms. There are axiomatic characterizations of several social choice and ranking rules that say that a given rule is the only one satisfying a particular set of axioms. Arrow’s impossibility theorem in social choice theory tells us that a certain set of, at first sight, quite reasonable axioms for a social ranking rule turns out to be inconsistent. Consequently, a social ranking rule that satisfies the axioms in question cannot exist. Finally, many notions from game theory, such as the Shapley–Shubik and the Banzhaf index, may also be characterized by a set of axioms. Full article

(This article belongs to the Special Issue Algebra, Logic and Applications)

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