Regular semigroups weakly generated by idempotents

Abstract

A regular semigroup is weakly generated by a set X if it has no proper regular subsemigroups containing X. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup FI(X) weakly generated by |X| idempotents such that all other regular semigroups weakly generated by |X| idempotents are homomorphic images of FI(X). The semigroup FI(X) is defined by a presentation G(X), ρe ∪ ρs and its structure is studied. Although each of the sets G(X), ρe, and ρs is infinite for |X| ≥ 2, we show that the word problem is decidable as each congruence class has a "canonical form". If FIn denotes FI(X) for |X| = n, we prove also that FI2 contains copies of all FIn 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 FI2; and (ii) all finite semigroups divide FI2.