On the reducibility order between borel equivalence relations

Abstract

This chapter focuses on the reducibility order between Borel equivalence relations. An equivalence relation E on a set X is a Borel equivalence relation if both X and E are Borel, in some Polish space, and its square, respectively. The class of all Borel equivalence relations is denoted by BOREQ. This defines a quasi-ordering ≤ on BOREQ, with associated equivalence ≡. The chapter describes the Friedman-Stanley theorem for all E in BOREQ with at least two classes, E < E+. It is noted that BOREQ, ≤ has no maximum element. The proof of the theorem brings in an interesting invariant of the reducibility equivalence relation ≡, the potential Wadge class of a Borel equivalence relation. A jump operator in BOREQ is also discussed in the chapter.