ON WEAKLY PERFECT ANNIHILATING-IDEAL GRAPHS

2021 ◽  
pp. 1-11
Author(s):  
GANESH S. KADU ◽  
VINAYAK JOSHI ◽  
SAMRUDDHA GONDE
Keyword(s):  
Large Class ◽  
Commutative Semigroup ◽  
Zero Divisor ◽  
Commutative Semigroups

Abstract We prove that the annihilating-ideal graph of a commutative semigroup with unity is, in general, not weakly perfect. This settles the conjecture of DeMeyer and Schneider [‘The annihilating-ideal graph of commutative semigroups’, J. Algebra469 (2017), 402–420]. Further, we prove that the zero-divisor graphs of semigroups with respect to semiprime ideals are weakly perfect. This enables us to produce a large class of examples of weakly perfect zero-divisor graphs from a fixed semigroup by choosing different semiprime ideals.

scholarly journals Commutative semigroups whose endomorphisms are power functions

2021 ◽  
Author(s):  
Ryszard Mazurek
Keyword(s):  
Positive Integer ◽  
Power Function ◽  
Cyclic Group ◽  
Commutative Semigroup ◽  
Commutative Monoid ◽  
Power Functions ◽  
Commutative Semigroups ◽  
Finite Cyclic Group

AbstractFor any commutative semigroup S and positive integer m the power function $$f: S \rightarrow S$$ f : S → S defined by $$f(x) = x^m$$ f ( x ) = x m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is 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. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

scholarly journals The regular radical of semigroup rings of commutative semigroups

1992 ◽  
Vol 34 (2) ◽  
pp. 133-141 ◽  
Author(s):  
A. V. Kelarev
Keyword(s):  
Regular Semigroup ◽  
Commutative Semigroup ◽  
General Problem ◽  
Associative Ring ◽  
Complete Solution ◽  
Group Rings ◽  
Semigroup Rings ◽  
Regular Group ◽  
Commutative Semigroups

A description of regular group rings is well known (see [12]). Various authors have considered regular semigroup rings (see [17], [8], [10], [11], [4]). These rings have been characterized for many important classes of semigroups, although the general problem turns out to be rather difficult and still has not got a complete solution. It seems natural to describe the regular radical in semigroup rings for semigroups of the classes mentioned. In [10], the regular semigroup rings of commutative semigroups were described. The aim of the present paper is to characterize the regular radical ρ(R[S]) for each associative ring R and commutative semigroup S.

Bases for commutative semigroups and groups

2008 ◽  
Vol 145 (3) ◽  
pp. 579-586 ◽  
Author(s):  
NEIL HINDMAN ◽  
DONA STRAUSS
Keyword(s):  
Abelian Group ◽  
Finite Order ◽  
Commutative Semigroup ◽  
Finite Subset ◽  
Surprising Result ◽  
Commutative Semigroups ◽  
Or Groups

AbstractA base for a commutative semigroup (S, +) is an indexed set 〈xt〉t∈A in S such that each element x ∈ S is uniquely representable as Σt∈Fxt where F is a finite subset of A and, if S has an identity 0, then 0 = Σn∈Øxt. We investigate those commutative semigroups or groups which have a base. We obtain the surprising result that has a base. More generally, we show that an abelian group has a base if and only if it has no elements of odd finite order.

On the Jacobson radical of semigroup rings of commutative semigroups

1990 ◽  
Vol 108 (3) ◽  
pp. 429-433 ◽  
Author(s):  
A. V. Kelarev
Keyword(s):  
Commutative Semigroup ◽  
Jacobson Radical ◽  
Associative Ring ◽  
Complete Solution ◽  
Semigroup Rings ◽  
Commutative Semigroups ◽  
Free Commutative Semigroup

Many authors have considered the radicals of semigroup rings of commutative semigroups. A list of the papers pertaining to this field is contained in [4]. In [1] Amitsur proved that, for any associative ring R and for every free commutative semigroup S, the equalities B(RS) = B(R)S and L(RS) = L(R)S hold, where B is the Baer radical and L is the Levitsky radical. A natural problem which arises is to describe semigroup rings RS such that π(RS) = π(R)S, where π is one of the most important radicals. For the Baer and Levitsky radicals and commutative semigroups a complete solution of the above problem follows from theorems 2·8 and 3·1 of [15].

Commutative Zero-divisor Semigroups of Graphs with at Most Four Vertices

2009 ◽  
Vol 16 (02) ◽  
pp. 341-350 ◽  
Author(s):  
Gaohua Tang ◽  
Huadong Su ◽  
Beishang Ren
Keyword(s):  
Commutative Semigroup ◽  
Zero Divisor ◽  
Simple Graphs ◽  
Zero Divisor Graph ◽  
Zero Divisors

The zero-divisor graph of a commutative semigroup with zero is a graph whose vertices are the nonzero zero-divisors of the semigroup, with two distinct vertices joined by an edge in case their product in the semigroup is zero. In this paper, we study commutative zero-divisor semigroups determined by graphs. We determine all corresponding zero-divisor semigroups of all simple graphs with at most four vertices.

scholarly journals Solution of Several Functional Equations on Nonunital Semigroups Using Wilson’s Functional Equations with Involution

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jaeyoung Chung ◽  
Prasanna K. Sahoo
Keyword(s):  
Commutative Semigroup ◽  
Functional Equations ◽  
Complex Numbers ◽  
Commutative Semigroups ◽  
General Solutions

LetSbe a nonunital commutative semigroup,σ:S→San involution, andCthe set of complex numbers. In this paper, first we determine the general solutionsf,g:S→Cof Wilson’s generalizations of d’Alembert’s functional equations  fx+y+fx+σy=2f(x)g(y)andfx+y+fx+σy=2g(x)f(y)on nonunital commutative semigroups, and then using the solutions of these equations we solve a number of other functional equations on more general domains.

scholarly journals A characterization of Commutative Semigroups

2021 ◽  
Vol 12 (3) ◽  
pp. 5150-5155
Author(s):  
Dr. D. Mrudula Devi Et. al.
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Commutative Semigroup ◽  
Zero Semigroup ◽  
Completely Regular Semigroup ◽  
Semi Group ◽  
Commutative Semigroups ◽  
Completely Regular ◽  
Left Regular

This paper deals with some results on commutative semigroups. We consider (s,.) is externally commutative right zero semigroup is regular if it is intra regular and (s,.) is externally commutative semigroup then every inverse semigroup  is u – inverse semigroup. We will also prove that if (S,.) is a H -  semigroup then weakly cancellative laws hold in H - semigroup. In one case we will take (S,.) is commutative left regular semi group and we will prove that (S,.) is ∏ - inverse semigroup. We will also consider (S,.) is commutative weakly balanced semigroup  and then prove every left (right) regular semigroup is weakly separate, quasi separate and separate. Additionally, if (S,.) is completely regular semigroup we will prove that (S,.) is permutable and weakly separtive. One a conclusing note we will show and prove some theorems related to permutable semigroups and GC commutative Semigroups.

scholarly journals Polynomial density of commutative semigroups

1993 ◽  
Vol 48 (1) ◽  
pp. 151-162 ◽  
Author(s):  
Andrzej Kisielewicz ◽  
Norbert Newrly
Keyword(s):  
Commutative Semigroup ◽  
Equational Theory ◽  
Full Description ◽  
Commutative Semigroups ◽  
Equational Theories

An algebra is said to be polynomially n−dense if all equational theories extending the equational theory of the algebra with constants have a relative base consisting of equations in no more than n variables. In this paper, we investigate polynomial density of commutative semigroups. In particular, we prove that, for n > 1, a commutative semigroup is (n − 1)-dense if and only if its subsemigroup consisting of all n−factor-products is either a monoid or a union of groups of a bounded order. Moreover, a commutative semigroup is 0-dense if and only if it is a bounded semilattice. For semilattices, we give a full description of the corresponding lattices of equational theories.

On semigroup algebras of cancellative commutative semigroups

1991 ◽  
Vol 118 (1-2) ◽  
pp. 1-3 ◽  
Author(s):  
A. V. Kelarev
Keyword(s):  
Commutative Semigroup ◽  
Semigroup Algebras ◽  
Commutative Semigroups ◽  
Hereditary Radical

SynopsisA cancellative commutative semigroup s and a hereditary radical ρ are constructed such that ρ is S-homogeneous but not S-normal. This answers a question which arose in the literature.

Radicals of semigroup rings of commutative semigroups

1986 ◽  
Vol 99 (3) ◽  
pp. 435-445 ◽  
Author(s):  
J. Okniński ◽  
P. Wauters
Keyword(s):  
Commutative Ring ◽  
Commutative Semigroup ◽  
Jacobson Radical ◽  
Prime Radical ◽  
Semigroup Rings ◽  
Commutative Semigroups

In this paper we determine radicals of semigroup rings R[S] where R is an associative, not necessarily commutative, ring and S is a commutative semigroup. We will restrict ourselves to the prime radical P, the Levitzki radical L and the Jacobson radical J. At the end we will also give a few comments on the Brown-McCoy radical U.

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