The Sandwich Sets And Idempotent Separating Congruences On Eventually Regular Semigroups

In this paper, we mainly study the sandwich sets and idempotent sepa-rating congruences on eventually regular semigroups. It is composed of four sections.The first section is a brief introduction and the basic concepts and nec-essary preliminaries.In the second section, we mainly study some properties of one-sided sand-wich sets and the idempotent neighborhood in an eventually regular semigroup. We prove that S(e,f)f is a left and eS(e,f) is a right zero semigroup, and give some sufficient and necessary conditions for S(e, f) to be a left (right) zero semigroup. We also show that the sandwich set is isomorphic to the di-rect product of a left zero semigroup and a right zero semigroup, so that the sandwich set is a rectangular band. And we give the equivalent characteriza-tion of the relation≤r and≤l.In the third section, we mainly study some properties of one-sided sand-wich sets and some properties of the sandwich sets in completely eventually regular semigroup. We prove that S(a)ar(a)a′is a left zero semigroup and a′S(a)a(a) is a right zero semigroup and the sandwich set is isomorphic to the direct product of a left zero semigroup and a right zero semigroup.In the fourth section, we characterize idempotent separating congruences and give the condition forδto be an maximum idempotent separating con-gruence on eventually orthodox semigroups. We prove that an idempotent separating congruence on an eventually orthodox semigroup concerning cen-tralizer CS(H) of H in S.