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 PG(P) is constructed from a projection algebra P, using the recent groupoid approach to regular ∗-semigroups. The assignment P → PG(P) is a left adjoint to the forgetful functor that maps a regular ∗-semigroup S to its projection algebra P(S). In fact, the category of projection algebras is coreflective in the category of regular ∗-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 ∗-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 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 𝒯ℒ_n is the free regular ∗-semigroup over its own projection algebra P(𝒯ℒ_n).