Analytic equivalence relations and bi-embeddability

Abstract. Louveau and Rosendal [5] have shown that the relation of biembeddability 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,...

• Type: Journal
• Year: 2011
• Where: JSYML
• Authors: Sy-David Friedman, Luca Motto Ros
• Keywords: Analytic Equivalence Relation, Countable Structures, Equivalence Relation, JSYML 2011