Natural extensions of embeddable semigroup actions

TL;DR

This paper develops a theory of natural extensions for continuous actions of countable embeddable semigroups, highlighting limitations and characterizations of such extensions, especially in non-reversible and compact cases, with implications for topological dynamics.

Contribution

It introduces a new framework for natural extensions of semigroup actions, characterizes when extensions exist, and clarifies the role of reversibility and embedding choices.

Findings

Not all surjective semigroup actions extend to group actions.

The free group on a semigroup always admits an extension.

Natural extensions only work well in compact, left reversible cases.

Abstract

Semigroup actions and their invertible extensions are discussed. First, we develop a theory of natural extensions for continuous actions of countable, embeddable semigroups. Second, we demonstrate that not every surjective such action of a semigroup, which embeds into a group and generates it, can be extended to an action of said group, and that this phenomenon is specific to non-reversible semigroups. Furthermore, we characterize the free group on a semigroup (the group together with the embedding) as the unique pair that always admits such an extension, showing that both the choice of the receiving group and the embedding are crucial for this construction. Next, we prove that the classical notion of a natural extension -- requiring all other invertible extensions to factor through it -- only works in the context of compact extensions of left reversible semigroup actions and fails outside of it, thus providing a characterization of left reversibility. We finish by briefly studying topological dynamical properties of the natural extension in the amenable case.