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 Ulam-Hyers Stability of Trigonometric Functional Equation with Involution

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Jaeyoung Chung ◽  
Chang-Kwon Choi ◽  
Jongjin Kim
Keyword(s):  
Functional Equation ◽  
Commutative Semigroup ◽  
Functional Inequality ◽  
Commutative Group ◽  
Complex Numbers ◽  
Ulam Stability ◽  
Real Numbers ◽  
General Solutions

LetSandGbe a commutative semigroup and a commutative group, respectively,CandR+the sets of complex numbers and nonnegative real numbers, respectively, andσ:S→Sorσ:G→Gan involution. In this paper, we first investigate general solutions of the functional equationf(x+σy)=f(x)g(y)-g(x)f(y)for allx,y∈S, wheref,g:S→C. We then prove the Hyers-Ulam stability of the functional equation; that is, we study the functional inequality|f(x+σy)-f(x)g(y)+g(x)f(y)|≤ψ(y)for allx,y∈G, wheref,g:G→Candψ:G→R+.

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 General Solutions of Two Quadratic Functional Equations of Pexider Type on Orthogonal Vectors

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Margherita Fochi
Keyword(s):  
Functional Equations ◽  
Inner Product ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Restricted Domain ◽  
Product Spaces ◽  
General Solutions ◽  
Inner Product Spaces ◽  
Restricted Domains

Based on the studies on the Hyers-Ulam stability and the orthogonal stability of some Pexider-quadratic functional equations, in this paper we find the general solutions of two quadratic functional equations of Pexider type. Both equations are studied in restricted domains: the first equation is studied on the restricted domain of the orthogonal vectors in the sense of Rätz, and the second equation is considered on the orthogonal vectors in the inner product spaces with the usual orthogonality.

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].

scholarly journals Stability of Exponential Functional Equations with Involutions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jaeyoung Chung ◽  
Soon-Yeong Chung
Keyword(s):  
Commutative Semigroup ◽  
Functional Equations ◽  
Exponential Functional ◽  
Not Otherwise Specified ◽  
Stability Of Functional Equations ◽  
The Stability

LetSbe a commutative semigroup if not otherwise specified andf:S→ℝ. In this paper we consider the stability of exponential functional equations|f(x+σ(y))-g(x)f(y)|≤ϕ(x)or ϕ(y),|f(x+σ(y))-f(x)g(y)|≤ϕ(x)orϕ(y)for allx,y∈Sand whereσ:S→Sis an involution. As main results we investigate bounded and unbounded functions satisfying the above inequalities. As consequences of our results we obtain the Ulam-Hyers stability of functional equations (Chung and Chang (in press); Chávez and Sahoo (2011); Houston and Sahoo (2008); Jung and Bae (2003)) and a generalized result of Albert and Baker (1982).

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.

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