Restriction and Ehresmann Semigroups

Inverse semigroups form a variety of unary semigroups, that is, semigroups equipped with an additional unary operation, in this case a 7→ a−1. The theory of inverse semigroups is perhaps the best developed within semigroup theory, and relies on two factors: an inverse semigroup S is regular, and has semilattice of idempotents. Three major approaches to the structure of inverse semigroups have emerged. Effectively, they each succeed in classifying inverse semigroups via groups (or groupoids) and semilattices (or partially ordered sets). These are (a) the Ehresmann-Schein-Nambooripad characterisation of inverse semigroups in terms of inductive groupoids, (b) Munn’s use of fundamental inverse semigroups and his construction of the semigroup TE from a semilattice E, and (c) McAlister’s results showing on the one hand that every inverse semigroup has a proper (E-unitary) cover, and on the other, determining the structure of proper inverse semigroups in terms of groups, semilattices and partially ordered sets. The aim of this article is to explain how the above techniques, which were developed to study inverse semigroups, may be adapted for certain classes of bi-unary semigroups. The classes we consider are those of restriction and Ehresmann semigroups. The common feature is that the semigroups in each class possess a semilattice of idempotents; however, there is no assumption of regularity. For Professor K.P. Shum on the occasion of his 70th birthday