FREE PROFINITE SEMIGROUPS OVER SOME CLASSES OF SEMIGROUPS LOCALLY IN ${\mathcal D} {\rm G}$

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