On epimorphisms and structurally regular semigroups

Shabir Ahanger,Sakeena Bano,Wajih Ashraf,Aftab Shah

doi:10.52547/cgasa.15.1.231

Abstract

In this paper we study epimorphisms, dominions and related properties for some classes of structurally (n,m)-regular semigroups for any pair (n,m) of positive integers. In Section 2, after a brief introduction of these semigroups, we prove that the class of structurallly (n,m)-generalized inverse semigroups is closed under morphic images. We then prove the main result of this section that the class of structurally (n,m)-generalized inverse semigroups is saturated and, thus, in the category of all semigroups, epimorphisms in this class are precisely surjective morphisms. Finally, in the last section, we prove that the variety of structurally (o, n)-left regular bands is saturated in the variety of structurally (o, k)-left regular bands for all positive integers k and n with 1 ≤ k ≤ n.

Highlights

In this paper we study epimorphisms, dominions and related properties for some classes of structurally (n, m)-regular semigroups for any pair (n, m) of positive integers
In any category C, one can verify that every surjective morphism is epi but the converse is not true in general
Epimorphisms are studied via dominions whose notion was first introduced by Isbell in [6]

Summary

Introduction

In this paper we study epimorphisms, dominions and related properties for some classes of structurally (n, m)-regular semigroups for any pair (n, m) of positive integers. ([7], Result 4) Let U be a subsemigroup of a semigroup S and Dom(U, S) = S. A semigroup S is said to be structurally regular if there exists some ordered pair (n, m) of non-negative integers such that S/θ(n, m) is regular.

Results

Conclusion