A topological approach to discrete restriction semigroups and their algebras

TL;DR

This paper develops a topological framework for discrete restriction semigroups using étale categories, generalizing inverse semigroup theory and establishing new algebraic and structural results.

Contribution

It introduces a universal category for restriction semigroups, extending inverse semigroup theorems and connecting semigroup algebras with convolution algebras of topological categories.

Findings

Embedding of restriction semigroups into Boolean restriction semigroups

Topological ESN-type theorem for restriction semigroups

Isomorphism between semigroup algebra and convolution algebra

Abstract

We introduce a general framework, based on \'etale 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 \'etale 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.