η-Simple semigroups without zero and η ∗-simple semigroups with a least non-zero idempotent

Roman S Gigoń

doi:10.1007/s00233-012-9408-0

Abstract

A semigroup S is called η-simple if S has no semilattice congruences except S×S. Tamura in (Semigroup Forum 24:77–82, 1982) studied η-simple semigroups with a unique idempotent. In the present paper we consider a more general situation, that is, we investigate η-simple semigroups (without zero) with a least idempotent. Moreover, we study η∗-simple semigroups with zero which contain a least non-zero idempotent.

Highlights

We say that a semigroup S is a semilattice if a2 = a, ab = ba for all a, b ∈ S
An ideal P of a semigroup S is called prime if the condition ab ∈ P implies that a ∈ P or b ∈ P for all a, b ∈ S
Proposition 2.1 Let S = S0 be an η-simple semigroup with a least idempotent

Summary

Introduction

Lemma 1.1 A semigroup S [with zero] is E[∗]-inversive if and only if every [nonzero] ideal of S contains some [non-zero] idempotent of S. Proposition 2.1 Let S = S0 be an η-simple semigroup with a least idempotent. S is an ideal extension of a group by an η-simple semigroup.

Results

Conclusion