Relatively exchangeable structures

TL;DR

This paper investigates the structure and representation of relatively exchangeable random relational structures, providing a general Aldous–Hoover-type representation under trivial definable closure and a more detailed local description under ultrahomogeneity and $n$-disjoint amalgamation.

Contribution

It extends the theory of exchangeable structures by characterizing relatively exchangeable structures with new representation results based on properties of the reference structure.

Findings

Aldous–Hoover-type representation for structures with trivial definable closure

Local dependence description under ultrahomogeneity and $n$-disjoint amalgamation

Framework for understanding automorphism-invariant random relational structures

Abstract

We study random relational structures that are \emph{relatively exchangeable}---that is, whose distributions are invariant under the automorphisms of a reference structure $\mathfrak{M}$. When $\mathfrak{M}$ has {\em trivial definable closure}, every relatively exchangeable structure satisfies a general Aldous--Hoover-type representation. If $\mathfrak{M}$ satisfies the stronger properties of {\em ultrahomogeneity} and {\em $n$-disjoint amalgamation property} ($n$-DAP) for every $n\geq1$, then relatively exchangeable structures have a more precise description whereby each component depends locally on $\mathfrak{M}$.