Formations and i-Fitting classes of inverse semigroups, congruences and languages

This article starts by relating formations of inverse semigroups, of congruences on inverse semigroups and of congruences on free inverse semigroups. Then, i-formations of idempotent separating congruences and of languages are defined and their classes are proved to be in bijection, and in bijection with f-formations of inverse semigroups. In the second part, the concepts of i-Fitting class of inverse semigroups, of congruences and of languages are introduced. Bijections between s-fitting classes of inverse semigroups, i-fitting classes of congruences and i-fitting classes of languages are presented. Of special relevance are results on formations and Fitting classes induced by the inverse case on the group case.