Congruences on the Lattice of Pseudovarieties of Finite Semigroups

Abstract

As a step in a study of the lattice ℒ(ℱ) of pseudovarieties of finite semigroups that attempts to take full advantage of the underlying lattice structure, a family of complete congruences is introduced on ℒ(ℱ). Such congruences provide a framework from which to study ℒ(ℱ) both locally and globally. Each is associated with a mapping of the form [Formula: see text] for some special class [Formula: see text] of finite semigroups. In some instances the class is itself a pseudovariety while in others the class will be defined in terms of certain congruences associated with Green's relations. The basic properties of these complete congruences are presented and some relations between certain operators associated with these congruences are obtained.