A groupoid approach to regular *-semigroups

A cornerstone of inverse semigroup theory is the ESN Theorem, which states that the category of inverse semigroups is isomorphic to the category of inductive groupoids. This was generalised to regular semigroups by Nambooripad in his legendary 1979 memoir. In this talk I will discuss recent joint work with P.A. Azeef Muhammed on the intermediate class of regular *-semigroups. These semigroups have an involution, but their idempotents need not commute; key examples include partition, Brauer and Temperley-Lieb monoids. The role of the biordered set of idempotents is played by certain unary algebras of projections, and various categorical structures built on them. Specifically, the category of all regular *-semigroups is isomorphic to the category of so-called chained projection groupoids. Among other things, our groupoid approach leads to natural constructions of the classical maximum fundamental regular *-semigroups, and also new free projection-generated regular *-semigroups.