A class of actions of inverse semigroups
The purpose of this paper is to use the theory of so-called ordered groupoids to give a unified account of some important constructions in inverse semigroup theory. The paper is the third of a trilogy written by the author on this theme, and does indeed draw upon results of one of its predecessors, both of which have appeared in the J. Algebra. The classical Munn representation of an inverse semigroup \(S\) is generalized to a so-called ordered representation of \(S\), the ingredients of which, \(\gamma\) a homomorphism from \(S\) into the inverse semigroup of all order isomorphisms between order ideals of a poset \(X\), and \(p\), an isotone surjection of \(X\) onto the set of all idempotents of \(S\), which together satisfy two axioms. Both the Wagner-Preston representation, and the effective actions of O'Carroll and McAlister are special cases. A one-to-one correspondence is established between ordered representations of an inverse semigroup \(S\) and special kinds of covers of \(S\). Applications include a full categorical account of the theory of idempotent pure congruences.
scientific article; zbMATH DE number 843880