A characterization of Commutative Semigroups

Dr D Mrudula Devi Et Al

doi:10.17762/turcomat.v12i3.2065

Abstract

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.

Highlights

Research on commutative semigroup has a long history
Lawson (1996) made a good case that the earliest article which would currently receive a classification in an 1826 paper by Abel which clearly contains cancellative commutative semigroups
A. semigroup s is commutative if the defining binary operation is commutative

Summary

Introduction

Research on commutative semigroup has a long history. Lawson (1996) made a good case that the earliest article which would currently receive a classification in an 1826 paper by Abel which clearly contains cancellative commutative semigroups. In this paper we present the results on Commutative semigroups. Preliminaries 1.1.Definition: A semigroup (S, .) is Intra regular i.e., xa2y = a (or) ya2x = a. If x,y S, u, v, S and a positive integer n s.t. xn = uy and yn = vx. 1.3 .Definition: A semi group (S..) is said to be - Regular. If positive integer an = anxan a, x S and n is any. Definition: A semi group (S..) is said to be left - inverse semigroup if it is - regular and a = axa = aya ax = ay for all a, x, y S ( xa = ya)

Theorem

1.14. Theorem

1.15. Theorem