Decomposition of the lattice of pseudovarieties of finite semigroups induced by bands

Abstract

It has been shown by the authors that the mapping χ : V −→ V ∩ B, where B is the pseudovariety of finite bands, is a complete retraction of the lattice L(F) of pseudovarieties of finite semigroups onto the lattice of pseudovarieties of bands. It follows that the classes of the induced congruence χ on L(F), or on the lattice of subpseudovarieties L(W) for any subpseudovariety W of F, are intervals. In this paper we solve the membership problem for the upper limit of the classes of χ restricted to L(W) for various W ∈ L(F), including F itself, and provide bases of pseudoidentities for certain cases.

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

“…Now we are in a position to describe how the results of the present paper relate to the previously known results due to Reilly and Zhang [26] and to the authors (unpublished).…”

“…We think that idempotents of K A with |A| > 1 may also play a distinguished role in the theory. Here we shall prove that such idempotents are useful for constructing pro-identity bases for pseudovarieties of the form H and for studying identities of finite full transformation semigroups; further applications may be found in [26].…”

“…The Reilly-Zhang idempotent ρ is computable -this was mentioned in [26] though no algorithm to compute ρ was described in that paper. We reproduce here such an algorithm following a private communication by Norman Reilly.…”

“…We note that, as it was not clear if one can evaluate each of the words w n in polynomial time, the paper [26] left open the question of whether or not a polynomially computable idempotent in K A exists.…”

“…On the other hand, we observe in Section 3 that the initial problem easily reduces to constructing an idempotent in the least closed ideal of the free profinite semigroup with a suitable number of generators. Then we present such a construction along with a construction that has been independently found by Reilly and Zhang [26]. In Section 4 we describe a polynomial time algorithm calculating the values of the operation induced by the Reilly-Zhang idempotent in any given finite semigroup.…”