A groupoid approach to regular *-semigroups

James East – 17 January 2024, 17 January 2024

A cornerstone of inverse semigroup theory is the Ehresmann--Schein--Nambooripad 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 pair of talks we will discuss our recent work 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.

The first talk (James) will develop the theory, and focus on the main result of our recent paper (Adv Math, 2024): the category of regular *-semigroups is isomorphic to the category of (what we call) chained projection groupoids.

The second talk (Azeef) will cover an interesting consequence, namely the existence of free (idempotent- and projection-generated) regular *-semigroups. Specifically, the forgetful functor RSS -> PA (where these are the categories of regular *-semigroups and projection algebras) has a left adjoint PA -> RSS. At the object level, the adjoint constructs a so-called chain semigroup C_P from a projection algebra P, wherein C_P is realised as a quotient of a free category arising from P. This utilises what we call linked pairs of projections (which are analogous to Nambooripad's singular squares in biordered sets). We shall also glimpse the special cases of various diagram monoids, and if time permits we will discuss recent and ongoing joint work with Robert Gray (East Anglia) and Nik Ruskuc (St Andrews) in this direction.