Relatively exchangeable 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 $\mathfrak{M}$.