Journal of Symbolic Logic

Analytic equivalence relations and bi-embeddability

Sy-David Friedman and Luca Motto Ros

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Louveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of ℒω1ω) is far from complete (see [5, 2]).

In this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivalence relations which are complete under Borel reducibility, but in fact any analytic equivalence relation is Borel equivalent to such a relation. This result and the techniques introduced answer questions raised in [5] about the comparison between isomorphism and bi-embeddability. Finally, as in [5] our results apply not only to classes of countable structures defined by sentences of ℒω1ω, but also to discrete metric or ultrametric Polish spaces, compact metrizable topological spaces and separable Banach spaces, with various notions of embeddability appropriate for these classes, as well as to actions of Polish monoids.

Article information

J. Symbolic Logic, Volume 76, Issue 1 (2011), 243-266.

First available in Project Euclid: 4 January 2011

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E15: Descriptive set theory [See also 28A05, 54H05]

Analytic equivalence relation analytic quasi-order bi-embeddability Borel reducibility


Friedman, Sy-David; Motto Ros, Luca. Analytic equivalence relations and bi-embeddability. J. Symbolic Logic 76 (2011), no. 1, 243--266. doi:10.2178/jsl/1294170999.

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