It has long been known that the set of projections of a regular *-semigroup can be given the structure of a unary algebra. This gives rise to a forgetful functor from the category of regular *-semigroups to the category of projection algebras. It turns out that this functor has a left adjoint, which maps a projection algebra to an associated free (projection-generated) regular *-semigroup. The free semigroup is built from a natural ‘chain groupoid’, which can be understood topologically as the fundamental groupoid of a certain simplicial 2-complex.
In this talk I will outline the above theory, and discuss some results and consequences. For example, it turns out that Temperley-Lieb monoids are themselves free regular *-semigroups (over their own projection algebras), and are unique among diagram monoids in this regard. For another example, one can start with the projection algebra of an ‘adjacency semigroup’ (in the sense of Jackson and Volkov), and the free functor produces a new kind of graph semigroup.
This is all joint work with Bob Gray, Azeef Muhammed PA and Nik Ruškuc.