Equivalence structures and isomorphisms in the difference hierarchy

We examine the notion of structures and functions in the Ershov difference hierarchy, and of equivalence structures in particular. A equivalence structure A = (A,E) has universe A = ω and an equivalence relation E. The equivalence class [a] of a ∈ A is {b ∈ A : aEb} and the character K of A is {〈k, n〉 ∈ (ω − {0}) : A has at least n classes of size k}. It is known that for any character K, there exists an equivalence structure with character K if and only if K is Σ2 but that there exists a ∆ 0 2 character such that any equivalence structure with character K must have infinite equivalence classes. We show: (1) for any n-c.e. character K, there is an equivalence structure with character K and no infinite equivalence classes; (2) there is an ω-c.e. character K such that any equivalence structure with character K must have infinite equivalence classes; (3) For any ∆2 character K, there exists a d.c.e equivalence structure with no infinite equivalence classes and character K. We define the notions of α-c.e. functions and graph-α-c.e. functions and show: (1) Any nonempty Σ2 set is the range of 2-c.e. function; (2) for every n, there is an (n + 1)-c.e. function which is not graph-n-c.e.; (3) there is a graph-2-c.e function that is not ω-c.e.; (4) there is a 2-c.e. bijection such that f−1 is not ω-c.e. We define the notions of (weakly) α-c.e. and of graph-α-c.e. isomorphisms and show: (1) For each n, there exist computable equivalence structures which are n + 1-c.e. isomorphic but not weakly n-c.e. isomorphic; (2) there are computable equivalence structures which are graph-2-c.e isomorphic but not weakly ω-c.e. isomorphic. We show that a computable equivalence structure is computably categorical if and only if it is weakly ω-c.e. categorical, by examining all cases. We show that any computable equivalence structure with bounded character K (and any number of infinite equivalence classes) is relatively graph-2-c.e. categorical and we show that any computable equivalence structure with a finite number of infinite equivalence classes is relatively graph-ω-c.e. categorical. It follows that a computable equivalence structure is ∆2 categorical if and only if it is graph-ω-c.e. categorical.