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On the complexity of the relations of isomorphism and bi-embeddability
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On the complexity of the relations of isomorphism and bi-embeddability
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On the complexity of the relations of isomorphism and bi-embeddability
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Journal Article
On the complexity of the relations of isomorphism and bi-embeddability
2012
Overview
Given an Lω1ω\\mathcal {L}_{\\omega _1\\omega }-elementary class C\\mathcal {C}, that is, the collection of the countable models of some Lω1ω\\mathcal {L}_{\\omega _1 \\omega }-sentence, denote by ≅C\\cong _{\\mathcal {C}} and ≡C\\equiv _{\\mathcal {C}} the analytic equivalence relations of, respectively, isomorphisms and bi-embeddability on C\\mathcal {C}. Generalizing some questions of A. Louveau and C. Rosendal, in a paper by S. Friedman and L. Motto Ros they proposed the problem of determining which pairs of analytic equivalence relations (E,F)(E,F) can be realized (up to Borel bireducibility) as pairs of the form (≅C,≡C)(\\cong _{\\mathcal {C}}, \\equiv _{\\mathcal {C}}), C\\mathcal {C} some Lω1ω\\mathcal {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 a problem: under very mild conditions on EE and FF, it is always possible to find such an Lω1ω\\mathcal {L}_{\\omega _1\\omega }-elementary class C\\mathcal {C}.
Publisher
American Mathematical Society
