Beyond regular semigroups

The aim of this paper is to study weakly $$U$$U-regular semigroups, a wide class containing all regular semigroups and all abundant semigroups with a regular biordered set of idempotents. Here $$U$$U is a regular biordered set. To do this, we introduce the notions of an RBS category and a weakly regular category over a regular biordered set. We show that the category of weakly $$U$$U-regular semigroups and admissible morphisms is equivalent to the category of weakly regular categories and RBS functors. Our method arises from Nambooripad’s work on the connection between regular biordered sets and regular semigroups. However, there are completely different techniques, the first being the introduction of RBS categories and the second being that it is more convenient to investigate (RBS) categories equipped with pre-orders, rather than with partial orders. A special case of our work is the class of weakly $$U$$U-orthodox semigroups, that is, weakly $$U$$U-regular semigroups with $$U$$U a band, characterised in an earlier article by the author using generalised categories equipped with pre-orders. Our result may be regarded as an extension of Armstrong’s work on concordant semigroups in the abundant case.