Projection algebras and free projection- and idempotent-generated regular $*$-semigroups

TL;DR

The paper develops the theory of free projection-generated regular $*$-semigroups $PG(P)$ from a projection algebra $P$, via a groupoid approach that realizes regular $*$-semigroups as chained projection groupoids. It proves that the construction is left adjoint to the projection-functor and that the projection algebras form a coreflective subcategory of regular $*$-semigroups, yielding a robust freeness framework. Three presentations of $PG(P)$ are given: a direct $P$-based presentation and two quotients of the classical free semigroups IG$(E)$ and RIG$(E)$, connecting to biordered-set theory and topological models. The work also situates $PG(P)$ within a topological interpretation through the fundamental groupoids of graphs and complexes $G_P$, $K_P$, and $K_P'$, enabling a geometric lens on maximal subgroups and their structure. Examples including graph-based bridging path semigroups and the Temperley–Lieb monoids illustrate the reach and depth of the construction, linking diagrammatic algebras with free regular $*$-semigroups in a unifying framework.

Abstract

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 $P\mapsto PG(P)$ 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 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 $TL_n$ is the free regular $*$-semigroup over its own projection algebra $P(TL_n)$.

Figures (7)

• Figure 1: Diagrammatic representation, multiplication and involution in $\mathcal{P}_6$.
• Figure 2: A projection $p\in P$, and a $p$-linked pair $(e,f)$, as in Definition \ref{['defn:CP_P']}. Dotted and dashed lines indicate $\leq_{\mathrel{\mathscr F}}$ and $\leq$ relationships, respectively. Lines with arrows indicate the paths $\lambda(e,p,f)$ and $\rho(e,p,f)$. Of course, by Lemma \ref{['lem:epf']}\ref{['epf1']}, the dual diagram in which all four arrows are reversed is also valid. See Remark \ref{['rem:LPP']} for more details.
• Figure 3: The projections $e,f,p_i,u_i,v_i$ from the proof of Proposition \ref{['prop:PCnu']}, shown here in the case $k=4$. Dashed lines indicate $\leq$ relationships. Each arrow $s\to t$ represents the $P$-path $(s,t)\in\mathscr P$, so the upper and lower paths $e\to f$ represent $\lambda(e,\mathfrak c,f) = (e,u_1,\ldots,u_k,f)$ and $\rho(e,\mathfrak c,f) = (e,v_1,\ldots,v_k,f)$, respectively.
• Figure 4: Diamonds, triangles and degeneracy of linked pairs of projections; see Remark \ref{['rem:nondeg']}.
• Figure 5: The elements of the Motzkin monoid $\mathcal{M}_3$; see Example \ref{['eg:M3']}.

Theorems & Definitions (94)

• Theorem 2.3
• Lemma 3.4
• proof
• Remark 3.6
• Theorem 3.9: see EPA2024
• Definition 4.1
• Lemma 4.3
• proof
• Remark 4.4
• Remark 4.5