Invariantly universal analytic quasi-orders

Details Invariantly universal analytic quasi-orders Invariantly universal analytic quasi-orders

2010-03-25

arxiv.org/abs/1003.4932v2

Mathematics

Logic

31 pages, 1 figure, to appear in Transactions of the American Mathematical Society

Scientific paper

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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