Classification of $\omega$-categorical monadically stable structures

TL;DR

This paper classifies $ ext{omega}$-categorical monadically stable structures using automorphism groups, showing they are finitely interdefinable with finitely bounded homogeneous structures and have finitely many reducts.

Contribution

It provides a complete classification of $ ext{omega}$-categorical monadically stable structures and confirms Thomas' conjecture for this class.

Findings

Structures are interdefinable with finitely bounded homogeneous structures.

The class is minimal, closed under key operations, and characterized by automorphism groups.

Each structure has finitely many reducts up to interdefinability.

Abstract

A first-order structure $\mathfrak{A}$ is called monadically stable iff every expansion of $\mathfrak{A}$ by unary predicates is stable. In this article we give a classification of the class $\mathcal{M}$ of $\omega$-categorical monadically stable structures in terms of their automorphism groups. We prove in turn that $\mathcal{M}$ is smallest class of structures which contains the one-element pure set, closed under isomorphisms, and closed under taking finitely disjoint unions, infinite copies, and finite index first-order reducts. Using our classification we show that every structure in $\mathcal{M}$ is first-order interdefinable with a finitely bounded homogeneous structure. We also prove that every structure in $\mathcal{M}$ has finitely many reducts up to interdefinability, thereby confirming Thomas' conjecture for the class $\mathcal{M}$.