We consider countable Borel equivalence relations on quotient Borel spaces. We prove a generalization of the representation theorem of Feldman–Moore, but provide some examples showing that other very simple properties of countable equivalence relations on standard Borel spaces may fail in the context of nonsmooth quotients. In this paper we try to analyze some very basic questions about the str...

The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver ([19]) and Harrington-Kechris-Louveau ([5]) show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on ω is above equality on P(ω), the power set of ω, and any Borel equivalence relation strictly ...

We introduce and study the notion of functorial Borel complexity for Polish groupoids. Such a notion aims at measuring the complexity of classifying the objects of a category in a constructive and functorial way. In the particular case of principal groupoids such a notion coincide with the usual Borel complexity of equivalence relations. Our main result is that on one hand for Polish groupoids ...

We consider a descriptive version of Kakutani equivalence for Borel automorphisms of Polish spaces. Answering a question of Nadkarni, we show that up to this notion, there are exactly two Borel automorphisms: those which are smooth, and those which are not. Using this, we classify all Borel R-flows up to C∞ time-change isomorphism. We then extend the notion of Kakutani equivalence to all (not n...

An equigenerated monomial ideal [Formula: see text] in the polynomial ring is a Freiman if text], where analytic spread of and least number generators text]. In this paper, we classify certain classes Borel-type ideals, including Borel ideals with multiple principal text]-Borel which are Freiman.