Projection algebras and free projection- and idempotent-generated regular $\ast$-semigroups

Abstract

The purpose of this paper is to introduce a new family of semigroups - the free projection-generated regular $\ast$-semigroups - and initiate their systematic study. Such a semigroup $PG(P)$ is constructed from a projection algebra $P$, using the recent groupoid approach to regular $\ast$-semigroups. The assignment $P\mapsto PG(P)$ is a left adjoint to the forgetful functor that maps a regular $\ast$-semigroup $S$ to its projection algebra $P(S)$. In fact, the category of projection algebras is coreflective in the category of regular $\ast$-semigroups. The algebra $P(S)$ uniquely determines the biordered structure of the idempotents $E(S)$, up to isomorphism, and this leads to a category equivalence between projection algebras and regular $\ast$-biordered sets. As a consequence, $PG(P)$ can be viewed as a quotient of the classical free idempotent-generated (regular) semigroups $IG(E)$ and $RIG(E)$, where $E=E(PG(P))$; this is witnessed by a number of presentations in terms of generators and defining relations. The semigroup $PG(P)$ can also be interpreted topologically, through a natural link to the fundamental groupoid of a simplicial complex explicitly constructed from $P$. The theory is then 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 $\ast$-semigroup over its own projection algebra $P(T{L}_n)$.

48 pages, 7 figures, 4 tables. V2: incorporates referee's feedback, to appear in Adv Math