On Isomorphism Classes of computably Enumerable Equivalence Relations

We examine how degrees of computably enumerable equivalence relations (ceers) under computable reduction break down into isomorphism classes. Two ceers are isomorphic if there is a computable permutation of ω which reduces one to the other. As a method of focusing on nontrivial differences in isomorphism classes, we give special attention to weakly precomplete ceers. For any degree, we consider the number of isomorphism types contained in the degree and the number of isomorphism types of weakly precomplete ceers contained in the degree. We show that the number of isomorphism types must be 1 or ω, and it is 1 if and only if the ceer is self-full and has no computable classes. On the other hand, we show that the number of isomorphism types of weakly precomplete ceers contained in the degree can be any member of r0, ωs. In fact, for any n P r0, ωs, there is a degree d and weakly precomplete ceers E1, . . . , En in d so that any ceer R in d is isomorphic to Ei ‘D for some i ď n and D a ceer with domain either finite or ω comprised of finitely many computable classes. Thus, up to a trivial equivalence, the degree d splits into exactly n classes. We conclude by answering some lingering open questions from the literature: Gao and Gerdes [12] define the collection of essentially FC ceers to be those which are reducible to a ceer all of whose classes are finite. They show that the index set of essentially FC ceers is Π3-hard, though the definition is Σ4. We close the gap by showing that the index set is Σ 0 4-complete. They also use index sets to show that there is a ceer all of whose classes are computable, but which is not essentially FC, and they ask for an explicit construction, which we provide. Andrews and Sorbi [4] examined strong minimal covers of downwards-closed sets of degrees of ceers. We show that if pEiq is a uniform c.e. sequence of non-universal ceers, then t‘iďjEi | j P ωu has infinitely many incomparable strong minimal covers, which we use to answer some open questions from [4]. Lastly, we show that there exists an infinite antichain of weakly precomplete ceers.