On the complexity of the relations of isomorphism and bi-embeddability

Details On the complexity of the relations of isomorphism and bi-embeddability On the complexity of the relations of isomorphism and bi-embeddability

2011-12-01

arxiv.org/abs/1112.0349v1

Proceedings of the American Mathematical Society 140 (2012), no. 1, 309-323

Mathematics

Logic

15 pages

Scientific paper

Given an L_{\omega_1 \omega}-elementary class C, that is the collection of the countable models of some L_{\omega_1 \omega}-sentence, denote by \cong_C and \equiv_C the analytic equivalence relations of, respectively, isomorphism and bi-embeddability on C. Generalizing some questions of Louveau and Rosendal [LR05], in [FMR09] it was proposed the problem of determining which pairs of analytic equivalence relations (E,F) can be realized (up to Borel bireducibility) as pairs of the form (\cong_C,\equiv_C), C some L_{\omega_1 \omega}-elementary class (together with a partial answer for some specific cases). Here we will provide an almost complete solution to such problem: under very mild conditions on E and F, it is always possible to find such an L_{\omega_1 \omega}-elementary class C.

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