Profinite Identities for Finite Semigroups Whose Subgroups Belong to a Given Pseudovariety

Abstract

We introduce a series of new polynomially computable implicit operations on the class of all finite semigroups. These new operations enable us to construct a finite pro-identity basis for the pseudovariety H of all finite semigroups whose subgroups belong to a given finitely based pseudovariety H of finite groups.

Maximal subgroups of the minimal ideal of a free profinite monoid are free

“…The question was prompted by the discovery of free profinite subgroups by Almeida and Volkov [4], who subsequently characterized those free profinite subgroups which are retracts [5]. Recently, Almeida has shown that not all maximal subgroups of finitely generated free profinite monoids are free profinite groups, although he has provided a large class of examples that are free profinite [3].…”

The pseudovariety of semigroups of triangular matrices over a finite field

“…Now we return to our problem of a syntactic description of pseudovarieties of triangularizable semigroups. It is known (see [4,Section 2]) and easy to verify that the pseudovariety G p is defined by the pseudoidentity…”

E-local pseudovarieties

“…Another example is the pseudoidentity (x ω y ω ) ω = (y ω x ω ) ω which defines the pseudovariety BG. As a last example, Almeida and Volkov [3] showed that, if u i = v i , with i ∈ I, is a basis of pseudoidentities for a pseudovariety of groups H, then u In the last result, we identify all E-local pseudoidentities with only one variable. The elements of these equalities are elements of the free profinite semigroup on the alphabet with a unique element x, and so they can be written under the form x α , where α is an element of the profinite completion of (N, +).…”