Commutative Semigroups Research Articles

On the set of $\underline{Set}(n)$'s

The set of $\underline{\mathrm{Set}}(n)$'s for natural numbers $n$ is constructed. For this set it is proved that it is a commutative semi-group. The conditions for which it is a monoid are given.

Commutative nilpotent transformation semigroups

Cameron et al. determined the maximum size of a null subsemigroup of the full transformation semigroup T(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {T}(X)$$\\end{document} on a finite set X and provided a description of the null semigroups that achieve that size. In this paper we extend the results on null semigroups (which are commutative) to commutative nilpotent semigroups. Using a mixture of algebraic and combinatorial techniques, we show that, when X is finite, the maximum order of a commutative nilpotent subsemigroup of T(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {T}(X)$$\\end{document} is equal to the maximum order of a null subsemigroup of T(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {T}(X)$$\\end{document} and we prove that the largest commutative nilpotent subsemigroups of T(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {T}(X)$$\\end{document} are the null semigroups previously characterized by Cameron et al.

Arithmetics of flatness for monoids

Flatness in categories of modules over rings has been transferred to categories of acts over semigroups by Stenström in 1971. In this note we will focus on commutative cancellative semigroups with unit (“monoids”) and show that this notion is useful too in considering arithmetical questions. The notion of factorable (resp. pre-Schreier) modules is extended to acts over monoids. For example it is shown that these acts can be characterized by flatness.

Complemented zero-divisor graphs associated with finite commutative semigroups

Let S be a commutative semigroup with zero. The zero-divisor graph associated with S is the simple graph whose vertex set is given by the nonzero zero-divisors of S where two distinct vertices are adjacent if and only if their product is zero. A graph is called complemented if every vertex has an incident edge which is not the edge of any three cycle. Complemented zero-divisor graphs which are associated with a finite commutative semigroup are described in terms of their clique number. The algebraic properties of the finite semigroups S whose zero-divisor graphs are complemented are described in terms of the clique number of the graph. Infinite families of complemented zero-divisor graphs associated with a finite commutative semigroup are given.

Arithmetic of semisubtractive semidomains

A subset [Formula: see text] of an integral domain is called a semidomain if the pairs [Formula: see text] and [Formula: see text] are commutative and cancellative semigroups with identities. The multiplication of [Formula: see text] extends to the group of differences [Formula: see text], turning [Formula: see text] into an integral domain. In this paper, we study the arithmetic of semisubtractive semidomains (i.e. semidomains [Formula: see text] for which either [Formula: see text] or [Formula: see text] for every [Formula: see text]). Specifically, we provide necessary and sufficient conditions for a semisubtractive semidomain to be atomic, to satisfy the ascending chain condition on principals ideals, to be a bounded factorization semidomain, and to be a finite factorization semidomain, which are subsequent relaxations of the property of having unique factorizations. In addition, we present a characterization of factorial and half-factorial semisubtractive semidomains. Throughout the paper, we present examples to provide insight into the arithmetic aspects of semisubtractive semidomains.

Polynomial maps and polynomial sequences in groups

Abstract This paper presents a modified version of Leibman’s group-theoretic generalizations of the difference calculus for polynomial maps from nonempty commutative semigroups to groups, and proves that it has many desirable formal properties when the target group is locally nilpotent and also when the semigroup is the set of nonnegative integers. We will apply it to solve Waring’s problem for general discrete Heisenberg groups in a sequel to this paper.

Generalized Polynomials on Semigroups

Abstract This article has two main parts. In the first part we show that some of the basic theory of generalized polynomials on commutative semi-groups can be extended to all semigroups. In the second part we show that if a sub-semigroup S of a group G generates G in the sense that G = S · S −1, then a generalized polynomial on S with values in an Abelian group H can be extended to a generalized polynomial on G into H. Finally there is a short discussion of the extendability of exponential functions and generalized exponential polynomials.

Topological embeddings into transformation monoids

Abstract In this paper we consider the questions of which topological semigroups embed topologically into the full transformation monoid ℕ ℕ {\mathbb{N}^{\mathbb{N}}} or the symmetric inverse monoid I ℕ {I_{\mathbb{N}}} with their respective canonical Polish semigroup topologies. We characterise those topological semigroups that embed topologically into ℕ ℕ {\mathbb{N}^{\mathbb{N}}} and belong to any of the following classes: commutative semigroups, compact semigroups, groups, and certain Clifford semigroups. We prove analogous characterisations for topological inverse semigroups and I ℕ {I_{\mathbb{N}}} . We construct several examples of countable Polish topological semigroups that do not embed into ℕ ℕ {\mathbb{N}^{\mathbb{N}}} , which answer, in the negative, a recent open problem of Elliott et al. Additionally, we obtain two sufficient conditions for a topological Clifford semigroup to be metrizable, and prove that inversion is automatically continuous in every Clifford subsemigroup of ℕ ℕ {\mathbb{N}^{\mathbb{N}}} . The former complements recent works of Banakh et al.

On LA-Gamma-semihypergroups containing two-sided bases

Ideal theory and base theory of algebraic structures play an important role in studied. In this study, we focus on studying bases of ideals in algebraic structures. The algebraic structure LA-Γ-semihypergroups is very interesting to study. This structure is generalization of many algebraic structures, for example, commutative semigroups, commutative Γ-semigroups, LA-semigroups, LA-Γ-semigroups, commutative semihypergroups, etc. Many results in these algebraic structures were extended to results in LA-Γ-semihypergroups. On the other hand, the results in these structures are the special cases of results in LA-Γ-semihypergroups. The main purpose of this paper is to introduce the notion of two-sided base of an LA-Γ-semihypergroup with pure left identity. We prove that every two-sided base of an LA-Γ-semihypergroup with pure left identity is one-element base. Finally, we prove that the compliment of union of all two-sided base of an LA-Γ-semihypergroup with pure left identity is maximal proper Γ-hyperideal.

Absolutely closed semigroups

Let C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document} be a class of topological semigroups. A semigroup X is called absolutely C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document}-closed if for any homomorphism h:X→Y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h:X\\rightarrow Y$$\\end{document} to a topological semigroup Y∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Y\\in {\\mathcal {C}}$$\\end{document}, the image h[X] is closed in Y. Let T1S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {T}_{\\!\ extsf {1}}\ extsf {S}$$\\end{document}, T2S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {T}_{\\!\ extsf {2}}\ extsf {S}$$\\end{document}, and TzS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {T}_{\\!\ extsf {z}}\ extsf {S}$$\\end{document} be the classes of T1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_1$$\\end{document}, Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup X is absolutely TzS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {T}_{\\!\ extsf {z}}\ extsf {S}$$\\end{document}-closed if and only if X is absolutely T2S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {T}_{\\!\ extsf {2}}\ extsf {S}$$\\end{document}-closed if and only if X is chain-finite, bounded, group-finite and Clifford + finite. On the other hand, a commutative semigroup X is absolutely T1S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {T}_{\\!\ extsf {1}}\ extsf {S}$$\\end{document}-closed if and only if X is finite. Also, for a given absolutely C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document}-closed semigroup X we detect absolutely C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document}-closed subsemigroups in the center of X.

Universal lifting properties of abelian semigroup actions

In this paper, we prove that every action of a commutative cancellative semigroup can be extended to an invertible action in some kinds of categories, and this extension has universal lifting property. And the universal extension functor preserves all limits in the categorical point of view, especially, it preserves inverse limits and products. Moreover, we give some applications in dynamical systems.

$\hbar$-CENTRALIZERS ON PSEUDOQUOTIENTS

A space of pseudoquotients, denoted by $\mathcal{B}(X,S)$, is defined as equivalence classes of pairs $(x,f)$, where $x$ is an element of a non-empty set $X$, $f$ is an element of $S$, a commutative semigroup of injective maps from $X$ to $X$, and $(x,f) \sim (y,g)$ when $gx=fy$. If $X$ is a ring and elements of $S$ are ring homomorphisms, then $\mathcal{B}(X,S)$ is a ring. We show that, under natural conditions, a $\hbar$ -Jordan centralizer on $X$ has a unique extension to a $\hbar$-Jordan centralizer on $\mathcal{B}(X,S)$

Some Algebraic and Topological Structures of Laplace Transformable Functions

Abstract: In this work, the set of all functions that are Laplace transformable with regard to their structures both algebraic and topological, is taken into account. Certain topological properties of the set of Laplace transformable functions with the help of a metric are described. Also, we determine the proofs of the statements that the set of all Laplace transformable functions is a commutative semi-group with respect to the convolution operation as well as an Abelian group with respect to the operation of addition. Metric for two functions belonging to the set of all Laplace transformable functions is defined and the proof that the Laplace transformable functions' space is complete with our metric is given. The separability theorem and that the Laplace transformable functions' space is disconnected are also discussed. Keywords: Abelian group, Commutative semi-group, Disconnected space, Laplace transform, Separability theorem.

Free pre-Lie family algebras

In this paper, we first define the pre-Lie family algebra associated to a dendriform family algebra in the case of a commutative semigroup. Then we construct a pre-Lie family algebra via typed decorated rooted trees, and we prove the freeness of this pre-Lie family algebra. We also construct pre-Lie family operad in terms of typed labeled rooted trees, and we obtain that the operad of pre-Lie family algebras is isomorphic to the operad of typed labeled rooted trees, which generalizes the result of Chapoton and Livernet. In the end, we construct Zinbiel and pre-Poisson family algebras and generalize results of Aguiar.

On a representation of t-norms on bounded meet semilattices

We investigate how and when a t-norm on a bounded meet semilattice can be represented by a t-norm on some sub-semilattice and a collection of interior operators. We present such a representation using a novel notion of consistent triplet consisting of a t-norm, a commutative semigroup of interior operators, and a meet semilattice of symmetric lower sets. Furthermore, we study a related problem of extension of a t-norm from a sub-semilattice to the whole semilattice given a consistent triplet. This gives us a new construction method for t-norms on bounded semilattices that covers several known methods. Our results are illustrated by several examples.

Aichinger equation on commutative semigroups

We consider Aichinger?s equation f (x1 +... + xm+1) = Xm+1 i=1 1i(x1, x2, ...,bxi,..., xm+1) for functions defined on commutative semigroups which take values on commutative groups. The solutions of this equation are, under very mild hypotheses, generalized polynomials. We use the canonical form of generalized polynomials to prove that compositions and products of generalized polynomials are again generalized polynomials and that the bounds for the degrees are, in this new context, the natural ones. In some cases, we also show that a polynomial function defined on a semigroup can uniquely be extended to a polynomial function defined on a larger group. For example, if f solves Aichinger?s equation under the additional restriction that x1,..., xm+1 ? Rp +, then there exists a unique polynomial function F defined on Rp such that F|R p + = f . In particular, if f is also bounded on a set A ? Rp + with positive Lebesgue measure then its unique polynomial extension F is an ordinary polynomial in p variables with total degree ? m, and the functions 1i are also restrictions to Rpm + of ordinary polynomials of total degree ? m defined on Rpm.

Structure of relatively free n-tuple semigroups

An n-tuple semigroup is an algebra defined on a set with n binary associative operations. This notion was considered by Koreshkov in the context of the theory of n-tuple algebras of associative type. The n>1 pairwise interassociative semigroups give rise to an n-tuple semigroup. This paper is a survey of recent developments in the study of free objects in the variety of n-tuple semigroups. We present the constructions of the free n-tuple semigroup, the free commutative n-tuple semigroup, the free rectangular n-tuple semigroup, the free left (right) k-nilpotent n-tuple semigroup, the free k-nilpotent n-tuple semigroup, and the free weakly k-nilpotent n-tuple semigroup. Some of these results can be applied to constructing relatively free cubical trialgebras and doppelalgebras.

Calculation of $\Sigma$-function for commutative non-cyclic semigroup of order 3 without identity and zero elements

Calculation of $\Sigma$-function for commutative non-cyclic semigroup of order 3 without identity and zero elements

Special noncancellative commutative semigroup operations on the real numbers and distribution

Special noncancellative commutative semigroup operations on the real numbers and distribution

Categorically Closed Unipotent Semigroups

Let C be a class of T1 topological semigroups, containing all Hausdorff zero-dimensional topological semigroups. A semigroup X is C-closed if X is closed in any topological semigroup Y∈C that contains X as a discrete subsemigroup; X is injectively C-closed if for any injective homomorphism h:X→Y to a topological semigroup Y∈C the image h[X] is closed in Y. A semigroup X is unipotent if it contains a unique idempotent. It is proven that a unipotent commutative semigroup X is (injectively) C-closed if and only if X is bounded and nonsingular (and group-finite). This characterization implies that for every injectively C-closed unipotent semigroup X, the center Z(X) is injectively C-closed.