Relatively exchangeable structures. Crane, H. & Towsner, H. Journal of Symbolic Logic, 83(2):416-442, 2018.
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We study random relational structures that are \emphrelatively exchangeable—that is, whose distributions are invariant under the automorphisms of a reference structure 𝔐. When 𝔐 has trivial definable closure, every relatively exchangeable structure satisfies a general Aldous–Hoover-type representation. If 𝔐 satisfies the stronger properties of ultrahomogeneity and n-disjoint amalgamation property (n-DAP) for every n≥1, then relatively exchangeable structures have a more precise description whereby each component depends locally on 𝔐.
@ARTICLE{2015arXiv150906733C,
   author = {{Crane}, H. and {Towsner}, H.},
    title = "{Relatively exchangeable structures}",
journal={Journal of Symbolic Logic},
year = 2018,
volume={83},
number={2},
pages={416-442},
urlarxiv={http://arxiv.org/abs/1509.06733},
urljournal={https://doi.org/10.1017/jsl.2017.61},
doi={10.1017/jsl.2017.61},
abstract={We study random relational structures that are \emph{relatively exchangeable}---that is, whose distributions are invariant under the automorphisms of a reference structure &#x1D510;. When &#x1D510; has <i>trivial definable closure</i>, every relatively exchangeable structure satisfies a general Aldous--Hoover-type representation. If &#x1D510; satisfies the stronger properties of <i>ultrahomogeneity</i> and <i>n-disjoint amalgamation property</i> (n-DAP) for every n&geq;1, then relatively exchangeable structures have a more precise description whereby each component depends locally on &#x1D510;.},
}

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