Cofinal families of borel equivalence relations and quasiorders

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 omega 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-sigma complete.