Natural extensions of embeddable semigroup actions

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.