Variations on Wadge Reducibility Extended Abstract

Wadge reducibility in the Baire and Cantor spaces is very important in descriptive set theory. We consider Wadge reducibility in so called phi-spaces which are topological counterpart of the algebraic directed-complete partial orderings. It turns out that in many spaces the Wadge reducibility behaves worse than in the classical case but there exist also interesting examples of spaces with a better behaviour.