Cofinal families of Borel equivalence relations and quasiorders

Abstract

Families of Borel equivalence relations and quasiorders that are cofinal with respect to the Borel reducibility ordering, ≤_B, are constructed. There is an analytic ideal on ω generating a complete analytic equivalence relation and any Borel equivalence relation reduces to one generated by a Borel ideal. Several Borel equivalence relations, among them Lipschitz isomorphism of compact metric spaces, are shown to be K_σ complete.