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On Polynomial-Time Relation Reducibility

We study the notion of polynomial-time relation reducibility among computable equivalence relations. We identify some benchmark equivalence relations and show that the reducibility hierarchy has a rich structure. Specif- ically, we embed the partial order of all polynomial-time computable sets into the polynomial-time relation reducibility hierarchy between two benchmark equivalence relations El and id. In addition, we consider equivalence relations with finitely many non-trivial equivalence classes and those whose equivalence classes are all finite.