Invariantly universal analytic quasi-orders

Abstract

We introduce the notion of an invariantly universal pair ( S , E ) (S,E) where S S is an analytic quasi-order and E ⊆ S E\subseteq S is an analytic equivalence relation. This means that for any analytic quasi-order R R there is a Borel set B B invariant under E E such that R R is Borel bireducible with the restriction of S S to B B . We prove a general result giving a sufficient condition for invariant universality, and we demonstrate several applications of this theorem by showing that the phenomenon of invariant universality is widespread. In fact it occurs for a great number of complete analytic quasi-orders, arising in different areas of mathematics, when they are paired with natural equivalence relations.