Ultrahomogeneity and $\omega$-categoricity of monounary algebras

TL;DR

This paper classifies monounary algebras based on their ultrahomogeneity and $oldsymbol{ extomega}$-categoricity, linking these properties to automorphism groups and element heights.

Contribution

It provides a complete characterization of $oldsymbol{ extomega}$-categorical and ultrahomogeneous monounary algebras of any size, connecting model theory concepts with algebraic automorphism properties.

Findings

A monounary algebra is $oldsymbol{ extomega}$-categorical if every element has finite height.

A monounary algebra is ultrahomogeneous if automorphism group has finitely many 1-orbits.

Classification relates ultrahomogeneity to (partial)-homogeneity and transitivity.

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.