Classification of $\omega$-categorical monadically stable structures

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 $\mathcal{M}$.