Three Examples of Join Computations

Abstract

This article answers three questions of J. Almeida. Using combinatorial, algebraic and topological methods, we compute joins involving the pseudovariety of nite groups, the pseudovariety of semigroups in which each idempotent is a right zero and the pseudovariety generated by monoids M such that each idempotent of Mnf1g is a left zero. The need to organize nite semigroups into a hierarchy comes from several algorithmic problems in connection with computer science. The lattice of semigroup pseudovarieties (classes of nite semigroups closed under nite direct product, subsemigroup and homomorphic image) became the object of special consideration after the publication of Eilenberg’s treatise [11]. Many problems from language theory found indeed an interesting formulation within this scope. At the moment, one of the challenges is to understand some operators acting on pseudovarieties. In this perspective, topological approaches providing signican t results were developed during the last decade by Almeida. The present paper takes advantage of these techniques to answer three