Pontryagin duality between compact and discrete abelian inverse monoids

TL;DR

This paper establishes a duality theory for compact and discrete abelian inverse monoids, generalizing classical duality theorems for groups and semilattices, and characterizes reflexivity in terms of abelian structure and zero-dimensionality.

Contribution

It introduces a duality framework for inverse monoids that unifies existing dualities for groups and semilattices, providing new characterizations of reflexivity.

Findings

Reflexive inverse monoids are exactly abelian with zero-dimensional idempotent semilattices.

Dual inverse monoids of discrete (resp. compact) monoids are compact (resp. discrete).

Unification of Pontryagin-van Kampen and Hofmann-Mislove-Stralka dualities.

Abstract

For a topological monoid S the dual inverse monoid is the topological monoid of all identity preserving homomorphisms from S to the circle with attached zero. A topological monoid S is defined to be reflexive if the canonical homomorphism from S to its second dual inverse monoid is a topological isomorphism. We prove that a (compact or discrete) topological inverse monoid S is reflexive (if and) only if S is abelian and the idempotent semilattice of S is zero-dimensional. For a discrete (resp. compact) topological monoid its dual inverse monoid is compact (resp. discrete). These results unify the Pontryagin-van Kampen Duality Theorem for abelian groups and the Hofmann-Mislove-Stralka Duality Theorem for zero-dimensional topological semilattices.