Jumps of computably enumerable equivalence relations

Abstract

We study computably enumerable equivalence relations (or, ceers), under computable reducibility ≤, and the halting jump operation on ceers. We show that every jump is uniform join-irreducible, and thus join-irreducible. Therefore, the uniform join of two incomparable ceers is not equivalent to any jump. On the other hand there exist ceers that are not equivalent to jumps, but are uniform join-irreducible: in fact above any non-universal ceer there is a ceer which is not equivalent to a jump, and is uniform join-irreducible. We also study transfinite iterations of the jump operation. If a is an ordinal notation, and E is a ceer, then let E(a) denote the ceer obtained by transfinitely iterating the jump on E along the path of ordinal notations up to a. In contrast with what happens for the Turing jump and Turing reducibility, where if a set X is an upper bound for the A-arithmetical sets then X(2) computes A(ω), we show that there is a ceer R such that R≥Id(n), for every finite ordinal n, but, for all k, R(k)≱Id(ω) (here Id is the identity equivalence relation). We show that if a,b are notations of the same ordinal less than ω2, then E(a)≡E(b), but there are notations a,b of ω2 such that Id(a) and Id(b) are incomparable. Moreover, there is no non-universal ceer which is an upper bound for all the ceers of the form Id(a) where a is a notation for ω2.