Equivalence structures and isomorphisms in the difference hierarchy

Abstract

We examine the effective categoricity of equivalence structures via Ershov's difference hierarchy. We explore various kinds of categoricity available by distinguishing three different notions of isomorphism available in this hierarchy. We prove several results relating our notions of categoricity to computable equivalence relations: for example, we show that, for such relations, computable categoricity is equivalent to our notion of weak ω-c.e. categoricity, and that Δ⁰₂-categoricity is equivalent to our notion of graph-ω-c.e. categoricity.