2008 seminar talk: Analytic Equivalence Relations and Bi-Embeddability
Talk held by Luca Motto Ros (KGRC) at the KGRC seminar on 2008-12-04.
The analysis of the structure of analytic equivalence relations (ER for short) under Borel-reducibility has been one of the most relevant subject in the recent history of Descriptive Set Theory. Among ERs, a special place is occupied by the isomorphism relation on countable models of a certain Lω1,ω-sentence φ. However, isomorphism relations are just a special case, as there are many example of ERs which are even not Borel-reducible to an isomorphism relation. On the contrary, we will show in this talk that the bi-embeddability (resp. bi-homomorphism, bi-weak-homomorphism) relation is able to capture the whole complexity of the ER-structure: for every ER E there is an Lω1,ω-sentence φ such that E is Borel-equivalent to bi-embeddability (resp. bi-homomorphism, bi-weak-homomorphism) on the collection of countable models of φ. This is joint work with Sy D. Friedman.