No abstract

We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of F F -reducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all Σ 1 1 equivalence relations on hyperarithmetical subsets of ω.

The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver ([19]) and Harrington-Kechris-Louveau ([5]) show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on ω is above equality on P(ω), the power set of ω, and any Borel equivalence relation strictly above equality on the reals is above equality modulo finite on P(ω). In this article we examine the effective content of these and related results by studying effectively Borel equivalence relations under effectively Borel reducibility. The resulting structure is complex, even for equivalence relations with finitely many equivalence classes. However use of Kleene's O as a parameter is sufficient to restore the picture from the noneffective setting. A key lemma is the existence of two effectively Borel sets of reals, neither of which contains the range of the other under any effectively Borel function; the proof of this result applies Barwise compactness to a deep theorem of Harrington (see [6]) establishing for any recursive ordinal α the existence of Π 0 1 singletons whose α-jumps are Turing incomparable. * The authors acknowledge the generous support of the FWF through project P 19375-N18 ( * ) There are effectively Borel sets A and B such that for no effectively Borel function f does one have f [A] ⊆ B or f [B] ⊆ A.

No abstract

Let E be a computably enumerable (c.e.) equivalence relation on the set ω of natural numbers. We say that the quotient set ω/E (or equivalently, the relation E) realizes a linearly ordered set L if there exists a c.e. relation respecting E such that the induced structure (ω/E; ) is isomorphic to L. Thus, one can consider the class of all linearly ordered sets that are realized by ω/E; formally, K(E) = {L | the order-type L is realized by E}. In this paper we study the relationship between computability-theoretic properties of E and algebraic properties of linearly ordered sets realized by E. One can also define the following pre-order lo on the class of all c.e. equivalence relations: E 1 lo E 2 if every linear order realized by E 1 is also realized by E 2 . Following the tradition of computability theory, the lo-degrees are the classes of equivalence relations induced by the pre-order lo . We study the partially ordered set of lo-degrees. For instance, we construct various chains and antichains and show the existence of a maximal element among the lo-degrees.