Ultrahomogeneity and $\omega$-categoricity of monounary algebras

Abstract

Ultrahomogeneity and $\omega$-categoricity are two central concepts arising from model theory, with strong connections with oligomorphic permutation groups and quantifier elimination. In particular, both are conditions on the automorphism group of a structure. The aim of this paper is to describe both the $\omega$-categorical monounary algebras and the ultrahomogeneous monounary algebras of arbitrary cardinalities. We show that a monounary algebra is $\omega$-categorical [ultrahomogeneous] if and only if every element has finite height and Aut$(\mathcal{A})$ has only finitely many 1-orbits [$\mathcal{A}$ is 1-ultrahomogeneous]. Our classification of ultrahomogeneous monounary algebras is then viewed in the context of previously studied variants of ultrahomogeneity, including (partial)-homogeneity and transitivity.