A topological approach to discrete restriction semigroups and their algebras

TL;DR

This work develops a topological framework for discrete restriction semigroups by introducing the universal category ${\mathscr C}(S)$, built as germs of the spectral action on the spectrum $\widehat{P(S)}$. It shows that $S$ embeds into the Boolean restriction semigroup of compact slices ${\mathscr C}(S)^a$, enabling a topological ESN-type correspondence, a Petrich-Reilly-type structure theorem for proper restriction semigroups via partial actions, and a Steinberg-type isomorphism $KS \cong K{\mathscr C}(S)$ between semigroup algebras and convolution algebras of the universal category. The results extend the inverse-semigroup theory to the broader setting of restriction semigroups, including range and birestriction cases, and provide a unified, topological pathway to represent and analyze their algebras. The framework opens avenues for future work on operator algebras of étale categories and restriction semigroups, linking algebraic, categorical, and dynamical perspectives. Overall, the paper significantly broadens the scope of topological methods in semigroup theory and their algebraic incarnations.

Abstract

We introduce a general framework, based on étale topological categories, for studying discrete restriction semigroups and their algebras. Generalizing Paterson's universal groupoid of an inverse semigroup, we define the universal category ${\mathscr C}(S)$ of a restriction semigroup $S$ with local units as the category of germs of the spectral action of $S$ on the character space of its projection semilattice. This is an étale topological category, meaning that its domain map is a local homeomorphism, while its range map is only required to be continuous. We show that $S$ embeds into the universal Boolean restriction semigroup of compact slices of ${\mathscr C}(S)$ and apply this embedding to establish the following results: - a topological version of the ESN-type theorem for restriction semigroups by Gould and Hollings; - an extension to restriction semigroups of the Petrich-Reilly structure theorem for $E$-unitary inverse semigroups in terms of partial actions; - an isomorphism between the semigroup algebra of a restriction semigroup $S$ with local units and the convolution algebra of the universal category ${\mathscr C}(S)$, extending the seminal result by Steinberg. The paper is inspired by the work of Cockett and Garner and builds upon the earlier research of the author. It shows that the theory of restriction semigroups can be developed much further than was previously thought, as a natural extension of the inverse semigroup theory.

Theorems & Definitions (146)

• Theorem 1.1
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Example 2.5
• Definition 2.6
• Lemma 2.7
• Lemma 2.8
• proof