Regular semigroups, fundamental semigroups and groups

Abstract

In this paper we obtain necessary and sufficient conditions on a regular semigroup in order that it should be an idempotent separating homomorphic image of a full subsemigroup of the direct product of a group and a fundamental or combinatorial regular semigroup. The main tool used is the concept of a prehomomorphism 6: S -> T between regular semigroups. This is a mapping such that (ab) 9 ^a0b9 in the natural partial order on T.

Semigroups whose idempotents form a subsemigroup

“…Rhodes' "strong t y p e -/ / conjecture" is that S c -5 / / . For completeness we close this section by showing the connection with a paper of McAlister [17]. McAlister derives structure theorems for arbitrary regular semigroups S in terms of groups, fundamental regular semigroups, and CIG(S), (=the conjugate closure of the idempotents).…”

An Introduction to Covers for Semigroups

“…Thus we have the following result which was first made explicit by Trotter in [43]. Of course, our proof gives an infinite cover, but one of the two proofs in [43] (based on results in [29]) gives a finite regular cover for a finite regular semigroup S. It is perhaps worth mentioning that, in the regular case, our construction of a cover can be modified by using the set V (a) of inverses of a rather than the set W (a) of weak inverses of a.…”

Ordered groupoids and the holomorph of an inverse semigroup

“…The property that a −1 θ = (aθ) −1 was included in the original definition of a premorphism in [11]: its redundancy was noted in [12]. Proposition 1.4 is stated as part of Theorem 3.1.5 in [10].…”