A Generalisation of the Munn Semigroup

TL;DR

This paper extends the classical Munn semigroup construction by using presheaves over meet-semilattices, providing a broader framework that characterizes étale actions of inverse semigroups.

Contribution

It introduces the generalized Munn semigroup via presheaves, unifying and extending previous constructions by Zhitomirskiy and Reilly.

Findings

Generalized Munn semigroup construction using presheaves.

Idempotent-separating representations characterize étale actions.

Unification of previous Munn semigroup generalizations.

Abstract

To each meet-semilattice $E$ is associated an inverse semigroup $T_{E}$ called the Munn semigroup of $E$. We generalise this construction by replacing the meet-semilattice $E$ by a presheaf of sets $X$ over a meet-semilattice. The inverse semigroup $T_{X}$ that results is called the generalised Munn semigroup. Our construction can be viewed as a generalisation of one due to Zhitomirskiy as well as a restriction of one due to Reilly. We prove that idempotent-separating representations in to the generalised Munn semigroup characterise \'etale actions of inverse semigroups.