Boolean inverse semigroups and their type monoids

TL;DR

The paper surveys Boolean inverse semigroups and their type monoids, emphasizing techniques, direct proofs, and connections to $V(K\langle S\rangle)$. It develops a self-contained treatment of semisimple and finite cases, provides a rook-matrix presentation of the type monoid, and establishes a canonical link between $\mathrm{Typ}(S)$ and $V(K\langle S\rangle)$, especially for locally matricial $S$. It further applies the framework to graph inverse semigroups, proving that $\mathrm{Typ}(\mathcal{B}_{tight}(I(\Gamma)))$ recovers the graph monoid $M_{\Gamma}$, with a chain of maps tying it to $V(L_K(\Gamma))$. The results connect noncommutative Stone duality, Leavitt path algebras, and Steinberg algebras, providing practical tools and proofs for researchers working with ample groupoids, $C^*$-algebras, and related algebraic structures.

Abstract

This is an expository paper which provides a quick introduction to Boolean inverse semigroups and their type monoids, with the emphasis on techniques and insights of the theory, and also treats the connection of the type monoid ${\mathrm{Typ}}(S)$ of a Boolean inverse semigroup $S$ with the monoid $V(K\langle S\rangle)$ of the ring $K\langle S\rangle$ assigned to $S$. We give original direct and simple proofs of some known results, such as the structure of semisimple Boolean inverse semigroups, the presentation of the type monoid by generalized rook matrices. We also prove that the type monoid of the tight Booleanization of a graph inverse semigroup is isomorphic to the graph monoid of this semigroup.

Theorems & Definitions (64)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Definition 2.5
• Proposition 2.6