Constructing inverse semigroups from category actions

Abstract

The theory in this paper was motivated by an example of an inverse semigroup important in Girard's ‘Geometry of interaction’ programme for linear logic. At one level, the theory is a refinement of the Wagner-Preston representation theorem; we show that every inverse semigroup is isomorphic to an inverse semigroup of all partial symmetries (of a specific type) of some structure. At another level, the theory unifies and completes two classical theories: the theory of bisimple inverse monoids created by Clifford and subsequently generalised to all inverse monoids by Leech; and the theory of 0-bisimple inverse semigroups due to Reilly and McAlister. Leech showed that inverse monoids could be described by means of a class of right cancellative categories, whereas Reilly and McAlister showed that 0-bisimple inverse semigroups could be described by means of generalised RP-systems. In this paper, we prove that every inverse semigroup can be constructed from a category acting on a set satisfying what we term the ‘orbit condition’.