Beyond orthodox semigroups

We introduce the notions of a generalised category and of an inductive generalised category over a band. Our purpose is to describe a class of semigroups which we name weakly B-orthodox. In doing so we produce a new approach to characterising orthodox semigroups, by using inductive generalised groupoids. Here B denotes a band of idempotents; we note that if B is a semilattice then a weakly B-orthodox semigroup is exactly an Ehresmann semigroup. Weakly B-orthodox semigroups are analogues of orthodox semigroups, where the relations (R) over tilde (B) and (L) over tilde (B) play the role that R and L take in the regular case. We show that the category of weakly B-orthodox semigroups and admissible morphisms is isomorphic to the category of inductive generalised categories over bands and pseudo-functors. Our approach is influenced by Nambooripad's work on the connection between biordered sets and regular semigroups. However, there are significant differences in strategy, the first being the introduction of generalised categories and the second being that it is more convenient to consider (generalised) categories equipped with pre-orders, rather than with partial orders. Our work may be regarded as extending a result of Lawson for Ehresmann semigroups. (c) 2012 Elsevier Inc. All rights reserved.