Idempotent-generated semigroups and pseudovarieties

Abstract

Abstract. The operator which constructs the pseudovariety generated by the idempotent-generated semigroups of a given pseudovariety is investigated. Several relevant examples of pseudovarieties generated by their idempotentgenerated elements are given as well as some properties of this operator. Particular attention is paid to the pseudovarieties in {J, R, L, DA} concerning this operator and their generator ranks and idempotent-generator ranks.

E-local pseudovarieties

“…• V E is the pseudovariety generated by the idempotent-generated semigroups of V. The last operator was introduced in Almeida and Moura [2] and we refer the reader to that paper as needed, but we opt to present here an easy lemma that will be used frequently in this paper: Lemma 2.1 (Almeida and Moura [2]). The operator E has the following properties, where V and W are arbitrary pseudovarieties:…”

“…In this subsection, we determine some pseudovarieties of the form V E . For that purpose, we also use the operator E which is studied in detail in [2].…”

“…This provides additional motivation for the calculation of (DS) E which remains an open problem (see [2]). …”

“…Since, by [2], the pseudovarieties R, L, J and DA satisfy the equality V E = V, it is easy to obtain the following results: Proof.…”

On the question of embedding a semigroup into a semiband

“…Recently, Almeida and Moura [1] studied the question of embedding a semigroup into a semiband inside some pseudovarieties of semigroups using some known embeddings, but also some profinite techniques. As an example, they were able to show that every finite semigroup S whose regular D-classes are aperiodic semigroups is embeddable into a semiband with the same characteristic using profinite techniques that do not give us an explicit semiband construction.…”