SEMIGROUPS AND ORDERED CATEGORIES .1. THE REDUCED CASE

We investigate the relationship which exists between certain classes of ordered small categories, introduced by Charles Ehresmann in the course of his work on local structures, and the class of U-semiabundant semigroups, first studied by El-Qallali and by de Barros. U-semiabundant semigroups embrace both the abundant and the regular semigroups-the former have been studied by Fountain and his students in a number of papers, the latter constitute a well-known and wellinvestigated branch of semigroup theory. One consequence of the ideas presented in this paper is that the work of Nambooripad on inductive groupoids and regular semigroups should not be seen as sui generis but as belonging to the general framework of the theory of ordered small categories. © 1991.