Free profinite semigroups over some classes of semigroups locally in DG

Abstract

This paper is concerned with the structure of semigroups of implicit operations on various subpseudovarieties V of DReG intersection with LDG, where DReG and DG are the pseudovarieties of all semigroups S in which each regular D-class is, respectively, a rectangular group and a group, and where LDG is the pseudovariety of semigroups locally in DG. As an application, we give a characterization of the variety of languages recognized by semigroups in V and derive some join decompositions of pseudovarieties.