Cross-connections of the singular transformation semigroup

Abstract

Cross-connection is a construction of regular semigroups using certain categories called normal categories which are abstractions of the partially ordered sets of principal left (right) ideals of a semigroup. We describe the cross-connections in the semigroup Sing(X) of all non-invertible transformations on a set X. The categories involved are characterized as the powerset category (X) and the category of partitions π(X). We describe these categories and show how a permutation on X gives rise to a cross-connection. Further, we prove that every cross-connection between them is induced by a permutation and construct the regular semigroups that arise from the cross-connections. We show that each of the cross-connection semigroups arising this way is isomorphic to Sing(X). We also describe the right reductive subsemigroups of Sing(X) with the category of principal left ideals isomorphic to (X). This study sheds light into the more general theory of cross-connections and also provides an alternate way of studying the structure of Sing(X). © 2018 World Scientific Publishing Company.