We consider the computably enumerable sets under the relation of Qreducibility. We first give several results comparing the upper semilattice of c.e. Q-degrees, 〈RQ,≤Q 〉, under this reducibility with the more familiar structure of the c.e. Turing degrees. In our final section, we use coding methods to show that the elementary theory of 〈RQ,≤Q 〉 is undecidable.

A partial group as defined in [3] is a semigroup S which satisfies the following axioms. (i) For every x ∈ S, there exists a (necessarily unique) element ex ∈ S, called the partial identity of x such that exx =xex =x and if yx =xy =x then ex y = yex = ex. (ii) For every x ∈ S, there exists a (necessarily unique) element x−1 ∈ S, called the partial inverse of x such that xx−1 = x−1x = ex and exx...

If S is a semilattice with operators, then there is an implicational theory Q such that the congruence lattice Con(S) is isomorphic to the lattice of all implicational theories containing Q. The author and Kira Adaricheva have shown that lattices of quasi-equational theories are isomorphic to congruence lattices of semilattices with operators [1]. That is, given a quasi-equational theory Q, the...

From CON(ZFC) we obtain: 1. CON(ZFC + 2" is arbitrarily large + there is a locally finite upper semilattice of size W2 which cannot be embedded into the Turing degrees as an upper semilattice). 2. CON(ZFC + 2" is arbitrarily large + there is a maximal independent set of Turing degrees of size Xl). Introduction. Let 6D denote the set of Turing degrees ordered under the usual Turing reducibility,...

In this paper, we investigate the quotient semilattice R/M of the r.e. degrees modulo the cappable degrees. We first prove the R/M counterpart of the Friedberg-Muchnik theorem. We then show that minimal elements and minimal pairs are not present in R/M. We end with a proof of the R/M counterpart to Sack's splitting theorem. 0. Introduction. The set of all r.e. degrees is made into an upper semi...

We study here the degree-theoretic structure of set-theoretical splittings of recursively enumerable r.e. sets into diierences of r.e. sets. As a corollary we deduce that the ordering of wttdegrees of unsolvability of diierences of r.e. sets is not a distributive semilattice and is not elementarily equivalent to the ordering of r.e. wttdegrees of unsolvability.