Restriction Semigroups and Inductive Constellations

Abstract

The Ehresmann–Schein–Nambooripad (ESN) Theorem, stating that the category of inverse semigroups and morphisms is isomorphic to the category of inductive groupoids and inductive functors, is a powerful tool in the study of inverse semigroups. Armstrong and Lawson have successively extended the ESN Theorem to the classes of ample, weakly ample, and weakly E-ample semigroups. A semigroup in any of these classes must contain a semilattice of idempotents, but need not be regular. It is significant here that these classes are each defined by a set of conditions and their left-right duals. Recently, a class of semigroups has come to the fore that is a one-sided version of the class of weakly E-ample semigroups. These semigroups appear in the literature under a number of names: in category theory they are known as restriction semigroups, the terminology we use here. We show that the category of restriction semigroups, together with appropriate morphisms, is isomorphic to a category of partial semigroups we dub inductive constellations, together with the appropriate notion of ordered map, which we call inductive radiant. We note that such objects have appeared outside of semigroup theory in the work of Exel. In a subsequent article we develop a theory of partial actions and expansions for inductive constellations, along the lines of that of Gilbert for inductive groupoids.