Projection algebras and free projection- and idempotent-generated regular ⁎-semigroups

The purpose of this paper is to introduce a new family of semigroups—the free projection-generated regular ⁎-semigroups—and initiate their systematic study. Such a semigroup $\text{PG}\left(P\right)$ is constructed from a projection algebra P, using the recent groupoid approach to regular ⁎-semigroups. The assignment $P\mapsto \text{PG}\left(P\right)$ is a left adjoint to the forgetful functor that maps a regular ⁎-semigroup S to its projection algebra $P\left(S\right)$. In fact, the category of projection algebras is coreflective in the category of regular ⁎-semigroups. The algebra $P\left(S\right)$ uniquely determines the biordered structure of the idempotents $E\left(S\right)$, up to isomorphism, and this leads to a category equivalence between projection algebras and regular ⁎-biordered sets. As a consequence, $\text{PG}\left(P\right)$ can be viewed as a quotient of the classical free idempotent-generated (regular) semigroups $\text{IG}\left(E\right)$ and $\text{RIG}\left(E\right)$, where $E=E\left(\text{PG}\right(P\left)\right)$; this is witnessed by a number of presentations in terms of generators and defining relations. The semigroup $\text{PG}\left(P\right)$ can also be interpreted topologically, through a natural link to the fundamental groupoid of a simplicial complex explicitly constructed from P. The above theory is illustrated on a number of examples. In one direction, the free construction applied to the projection algebras of adjacency semigroups yields a new family of graph-based path semigroups. In another, it turns out that, remarkably, the Temperley–Lieb monoid $T{L}_n$ is the free regular ⁎-semigroup over its own projection algebra $P\left(T{L}_n\right)$.