Invariantly universal analytic quasi-orders

Camerlo, Riccardo; Marcone, Alberto; Motto Ros, Luca

doi:10.1090/S0002-9947-2012-05618-2

Invariantly universal analytic quasi-orders
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by Riccardo Camerlo, Alberto Marcone and Luca Motto Ros PDF

Trans. Amer. Math. Soc. 365 (2013), 1901-1931 Request permission

Abstract:

We introduce the notion of an invariantly universal pair $(S,E)$ where $S$ is an analytic quasi-order and $E\subseteq S$ is an analytic equivalence relation. This means that for any analytic quasi-order $R$ there is a Borel set $B$ invariant under $E$ such that $R$ is Borel bireducible with the restriction of $S$ to $B$. 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.

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