E-local pseudovarieties

Abstract

Abstract. Generalizing a property of the pseudovariety of all aperiodic semigroups observed by Tilson, we call E-local a pseudovariety V which satisfies the following property: for a finite semigroup, the subsemigroup generated by its idempotents belongs to V if and only if so do the subsemigroups generated by the idempotents in each of its regular D-classes. In this paper, we present several sufficient or necessary conditions for a pseudovariety to be E-local or for a pseudoidentity to define an E-local pseudovariety. We also determine several examples of the smallest E-local pseudovariety containing a given pseudovariety.

Idempotent-generated semigroups and pseudovarieties

“…Since every regular D-class D of S is a group, it follows that E(D) is trivial and, therefore, E(D) ∈ J. Since J is E-local (see [9,Example 3.6]), we have E(S) ∈ J. Hence J ⊆ DH ⊆ EJ, where the first inclusion is trivial.…”

“…The notion of E-local pseudovariety, introduced in [9], enables us to determine (DO) E and (DH) E , as we see in the following examples. Recall that a pseudovariety V is E-local if it satisfies the following property: given S ∈ S, E(S) ∈ V if and only if E(D) ∈ V, for every regular D-class D of S.…”

“…The notion of E -local pseudovariety, introduced in [11], enables us to determine (DO) E and (DH) E , as we see in the following examples. Recall that a pseudovariety V is E -local if it satisfies the following property: In an attempt to identify the pseudovarieties that are generated by their idempotentgenerated elements, we present the results below and in § § 4 and 5.…”