Commutative Semigroups Research Articles

Rees quotients of numerical semigroups

We introduce a class of finite semigroups obtained by considering Rees quotients of numerical semigroups. Several natural questions concerning this class, as well as particular subclasses obtained by considering some special ideals, are answered while others remain open. We exhibit nice presentations for these semigroups and prove that the Rees quotients by ideals of \mathbb{N} , the positive integers under addition, constitute a set of generators for the pseudovariety of commutative and nilpotent semigroups.

On common eigenvectors for semigroups of matrices in tropical and traditional linear algebra

We prove the existence of a common eigenvector for commutative, nilpotent and quasinilpotent semigroups of matrices with complex or real nonnegative entries both in the conventional and tropical linear algebra.

Semigroups whose endomorphisms are power functions

For any commutative semigroup S and any positive integer m, the power function f:S→S defined by f(x)=x m is an endomorphism of S. In this paper we characterize finite cyclic semigroups as those finite commutative semigroups whose endomorphisms are power functions. We also prove that if S is a finite commutative semigroup with 1≠0, then every endomorphism of S preserving 1 and 0 is equal to a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. Immediate consequences of the results are, on the one hand, a characterization of commutative rings whose multiplicative endomorphisms are power functions given by Greg Oman in the paper (Semigroup Forum, 86 (2013), 272–278), and on the other hand, a partial solution of Problem 1 posed by Oman in the same paper.

Interassociativity on a free commutative semigroup

We describe all interassociates of a free commutative semigroup. Each interassociate of a free commutative semigroup is proved to be a variant of it or coincides with that semigroup. Conditions are found for the isomorphy of two variants of a free commutative semigroup.

A semilattice structure for Arens products

Let (A,⋅) be a Banach algebra. Then for n∈ℕ, A (2n) has 2 n Arens products. In this paper we study the relations between the Arens products on A (2n). Moreover, if P n (A) denotes the set of all Arens products on A (2n), for n∈ℕ, we show that $P(A)=\bigcup_{n=1}^{\infty} P_{n}(A)$ is a ∧-semilattice. Also, we study P(A) as an infinite commutative semigroup and P(A)−{⋅} as a free semigroup generated by two elements. Then we investigate amenability and weak amenability for their semigroup Banach algebras.

Irreducible semigroups of positive operators on Banach lattices

The classical Perron–Frobenius theory asserts that an irreducible matrix has cyclic peripheral spectrum and its spectral radius is an eigenvalue corresponding to a positive eigenvector. This was extended by Radjavi and Rosenthal to semigroups of matrices and of compact operators on -spaces. We extend this approach to operators on an arbitrary Banach lattice . We prove, in particular, that if is a commutative irreducible semigroup of positive operators on containing a compact operator then there exist positive disjoint vectors in such that every operator in acts as a positive scalar multiple of a permutation on . Compactness of may be replaced with the assumption that is peripherally Riesz, i.e. the peripheral spectrum of is separated from the rest of the spectrum and the corresponding spectral subspace is finite dimensional. Applying the results to the semigroup generated an irreducible peripherally Riesz operator , we show that is a cyclic permutation on , , and if for some in and then for some and . We also extend results of Drnovšek and Levin about peripheral spectra of irreducible operators.

Endomorphism monoids in varieties of commutative semigroups

A concrete category is almost universal if its class of non-constant morphisms contains an isomorphic copy of every category of algebras as a full subcategory. This paper characterizes almost universal varieties of commutative semigroups. As a consequence we obtain that for every infinite cardinal κ there exists a commutative semigroup of cardinality κ such that it has exactly two endomorphisms, the identity endomorphism and a single constant endomorphism.

SET-NORM EXHAUSTIVE SET MULTIFUNCTIONS

In this paper we present some properties of set-norm exhaustive set multifunctions and also of atoms and pseudo-atoms of set multifunctions taking values in the family of non-empty subsets of a commutative semigroup with unity.

On De Graaf spaces of pseudoquotients

A space of pseudoquotients $\mathcal{B}(X,S)$ 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 $(x,f) \sim (y,g)$ if $gx=fy$. In this note we consider a generalization of this construction where the assumption of commutativity of $S$ by Ore type conditions. As in the commutative case, $X$ can be identified with a subset of $\mathcal{B}(X,S)$ and $S$ can be extended to a group $G$ of bijections on $\mathcal{B}(X,S)$. We introduce a natural topology on $\mathcal{B}(X,S)$ and show that all elements of $G$ are homeomorphisms on $\mathcal{B}(X,S)$.

Semigroup actions on ordered groupoids

Abstract In this paper we prove that if S is a commutative semigroup acting on an ordered groupoid G, then there exists a commutative semigroup S̃ acting on the ordered groupoid G̃:=(G × S)/ρ̄ in such a way that G is embedded in G̃. Moreover, we prove that if a commutative semigroup S acts on an ordered groupoid G, and a commutative semigroup S̄ acts on an ordered groupoid Ḡ in such a way that G is embedded in S̄, then the ordered groupoid G̃ can be also embedded in Ḡ. We denote by ρ̄ the equivalence relation on G × S which is the intersection of the quasi-order ρ (on G × S) and its inverse ρ −1.

Soft subsets and soft product operations

Molodtsov’s soft set theory provides a general mathematical framework for dealing with uncertainty. It is known that soft subsets and soft equal relations are of vital importance in soft set theory. This paper aims to give a systematic study on several types of soft subsets and various soft equal relations induced by them. We give some equivalent characterizations of different soft subsets and endeavor to ascertain the interrelations among these notions, illustrated by a number of concrete examples. We also consider ontology-based soft sets and show that soft L-subsets generalize both soft M-subsets and ontology-based soft subsets.Moreover, by means of soft L-subsets and some related notions, we give a theoretical study concerning soft product operations such as ∧-products and ∨-products. We consummate some incomplete results concerning soft product operations existing in the literature, and investigate the algebraic properties of soft product operations in detail. Finally, we consider free soft algebras associated with soft product operations. It is shown that soft L-equal relations are congruence relations over free soft algebras and the corresponding quotient structures form commutative semigroups.

Iterative methods for solving equilibrium problems, variational inequalities and fixed points of nonexpansive semigroups

In this work, strong convergence theorems by the viscosity approximation method associated with Meir---Keeler contractions are established for solving fixed point problems of a nonexpansive semigroup, a system of equilibrium problems and variational inequality problems in a real Hilbert space. Further, applications related to commutative semigroup are obtained.

Presentation of semidirect product of monogenic semigroups

By studying the semigroup presented by π = � A, B | A n+1 = A, B m+1 = B, BA = A n−1 B, B m = A n � , for every positive integers m, n ≥ 3 we show that, for even values of m this is an appropriate presentation for the semidirect product of monogenic semigroup S = � a | a n+1 = aby the monogenic semigroup T = � b | b m+1 = b� . Moreover, for odd values of m it is a commutative semigroup of order (3+(−1) n )m 2 . The interests of presentability of prod- ucts of semigroups are quite useful in the study of more effective and intrinsic properties of the products (like as the computation of their characters and studying their Green J-classes). Our method of proof based on using the theoretic definitions and hereby we will deduced the presentation of the direct product of monogenic semigroups for every integers n, m ≥ 3. Mathematics Subject Classification: 20M30, 20E45

On finitely related semigroups

An algebraic structure is finitely related (has finite degree) if its term func- tions are determined by some finite set of finitary relations. We show that the fol- lowing finite semigroups are finitely related: commutative semigroups, 3-nilpotent monoids, regular bands, semigroups with a single idempotent, and Clifford semi- groups. Further we provide the first example of a semigroup that is not finitely related: the 6-element Brandt monoid. This answers a question by Davey, Jackson, Pitkethly, and Szabo from Davey et al. (Semigroup Forum, 83(1):89-122, 2011).

Properties of a set of idempotent elements of generalised inverse semigroups

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

Left Reductive Congruences on Semigroups

A semigroup S is called a left reductive semigroup if, for all elements a,b∈S, the assumption “xa=xb for all x∈S” implies a=b. A congruence α on a semigroup S is called a left reductive congruence if the factor semigroup S/α is left reductive. In this paper we deal with the left reductive congruences on semigroups. Let S be a semigroup and ϱ a congruence on S. Consider the sequence ϱ (0)⊆ϱ (1)⊆⋯⊆ϱ (n)⊆⋯ of congruences on S, where ϱ (0)=ϱ and, for an arbitrary non-negative integer n, ϱ (n+1) is defined by (a,b)∈ϱ (n+1) if and only if (xa,xb)∈ϱ (n) for all x∈S. We show that \(\bigcup_{i=0}^{\infty}\varrho^{(i)}\subseteq \mathit{lrc}(\varrho )\) for an arbitrary congruence ϱ on a semigroup S, where lrc(ϱ) denotes the least left reductive congruence on S containing ϱ. We focuse our attention on congruences ϱ on semigroups S for which the congruence \(\bigcup_{i=0}^{\infty}\varrho^{(i)}\) is left reductive. We prove that, for a congruence ϱ on a semigroup S, \(\bigcup_{i=0}^{\infty}\varrho^{(i)}\) is a left reductive congruence of S if and only if \(\bigcup_{i=0}^{\infty}\iota_{(S/\varrho)}^{(i)}\) is a left reductive congruence on the factor semigroup S/ϱ (here ι (S/ϱ) denotes the identity relation on S/ϱ). After proving some other results, we show that if S is a Noetherian semigroup (which means that the lattice of all congruences on S satisfies the ascending chain condition) or a semigroup in which S n =S n+1 is satisfied for some positive integer n then the universal relation on S is the only left reductive congruence on S if and only if S is an ideal extension of a left zero semigroup by a nilpotent semigroup. In particular, S is a commutative Noetherian semigroup in which the universal relation on S is the only left reductive congruence on S if and only if S is a finite commutative nilpotent semigroup.

Convergence results for the zero-finding problem and fixed points of nonexpansive semigroups and strict pseudocontractions

In this work, we establish strong convergence theorems for solving the fixed point problem of nonexpansive semigroups and strict pseudocontractions, and the zero-finding problem of maximal monotone operators in a Hilbert space. We further apply our result to the convex minimization problem and commutative semigroups. MSC: 47H09; 47H10

Fixed point properties for semigroups of nonlinear mappings and amenability

In this paper we study fixed point properties for semitopological semigroup of nonexpansive mappings on a bounded closed convex subset of a Banach space. We also study a Schauder fixed point property for a semitopological semigroup of continuous mappings on a compact convex subset of a separated locally convex space. Such semigroups properly include the class of extremely left amenable semitopological semigroups, the free commutative semigroup on one generator and the bicyclic semigroup S1=〈a,b:ab=1〉.

On 2-absorbing commutative semigroups and their applications to rings

A commutative ring R is called 2-absorbing (Badawi in Bull. Aust. Math. Soc. 75:417–429, 2007) if for arbitrary elements a,b,c∈R, abc=0 if and only if ab=0 or bc=0 or ac=0. In this paper we study this concept in a more general framework of commutative (multiplicative) semigroups with 0. The results obtained apply to many ring theoretic situations and make it possible to describe similarities and differences among some variants of the notion. We pay a particular attention to graded rings. We also show that a conjecture from (Anderson and Badawi in Commun. Algebra 39:1646–1672, 2011) concerning n-absorbing rings holds for rings with torsion-free additive groups.

Classification of finite commutative semigroups for which the inverse monoid of local automorphisms is permutable

We present a classification of finite commutative semigroups for which the inverse monoid of local automorphisms is permutable.