Regular semigroups weakly generated by idempotents

Abstract

A regular semigroup is weakly generated by a set [Formula: see text] if it has no proper regular subsemigroup containing [Formula: see text]. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup [Formula: see text] weakly generated by [Formula: see text] idempotents such that all other regular semigroups weakly generated by [Formula: see text] idempotents are homomorphic images of [Formula: see text]. The semigroup [Formula: see text] is defined by a presentation [Formula: see text] and its structure is studied. Although each of the sets [Formula: see text], [Formula: see text], and [Formula: see text] is infinite for [Formula: see text], we show that the word problem is decidable as each congruence class has a “canonical form”. If [Formula: see text] denotes [Formula: see text] for [Formula: see text], we prove also that [Formula: see text] contains copies of all [Formula: see text] as subsemigroups. As a consequence, we conclude that (i) all regular semigroups weakly generated by a finite set of idempotents, which include all finitely idempotent generated regular semigroups, strongly divide [Formula: see text]; and (ii) all finite semigroups divide [Formula: see text].