One-to-one partial right translations of a right cancellative semigroup

Abstract

A one-to-one partial right translation of a semigroup S is a one-to-one partial transformation p of S whose domain is a left ideal and which has the property that for each a E Ap, x E S, (xa)p = x(ap). The set 3 of all one-to-one partial right translations of S is an inverse semigroup which reflects many of the properties of S. The construction of 9 gives rise to a functor from a category of semigroups to the category of inverse semigroups. The first two sections deal with the effect of this functor on objects and morphisms respectively. In the third section, the functor is applied to show that the category of right cancellative monoids and, so-called, permissible homomorphisms is naturally equivalent to a category of O-simple inverse semigroups. The category of O-simple inverse semigroups is explicitly described. This result can be considered as an analog to Clifford’s theorem which describes the structure of a bisimple inverse monoid in terms of its right unit subsemigroup.