In  we have considered a wide class of “well-behaved” reducibilities for sets of reals. In this paper we continue with the study of Borel reducibilities by proving a dichotomy theorem for the degree-structures induced by good Borel reducibilities. This extends and improves the results of  allowing to deal with a larger class of notions of reduction (including, among others, the Baire class ξ functions).
"Baire reductions and good Borel reducibilities." J. Symbolic Logic 75 (1) 323 - 345, March 2010. https://doi.org/10.2178/jsl/1264433924