Trees, congruences and varieties of finite semigroups

Abstract

A classification scheme for regular languages or finite semigroups was proposed by Pin through tree hierarchies, a scheme related to the concatenation product, an operation on languages, and to the Schützenberger product, an operation on semigroups. Starting with a variety of finite semigroups (or pseudovariety of semigroups) V, a pseudovariety of semigroups ♦ u( V) is associated to each tree u. In this paper, starting with the congruence γ A generating a locally finite pseudovariety of semigroups V for the finite alphabet A, we construct a congruence  u ( γA) in such a way to generate ♦ u( V) for A. We give partial results on the problem of comparing the congruences  u ( γA) or the pseudovarieties ♦ u( V) . We also propose case studies of associating trees to semidirect or two-sided semidirect products of locally finite pseudovarieties.