From right PP monoids to restriction semigroups: a survey

Abstract

Left restriction semigroups are a class of semigroups which generalise inverse semigroups and which emerge very naturally from the study of partial transformations of a set.  Consequently, they have arisen in a variety of different contexts, under a range of names.  One of the various guises under which left restriction semigroups have appeared is that of weakly left E-ample semigroups, as studied by Fountain, Gomes, Gould and Lawson, amongst others.  In the present article, we will survey the historical development of the study of left restriction semigroups, from the `weakly left E-ample' perspective, and sketch out the basic aspects of their theory.