\(sQ_1\)-degrees of computably enumerable sets

Abstract

We show that the sQ-degree of a hypersimple set includes an infinite collection of \(sQ_1\)-degrees linearly ordered under \(\le _{sQ_1}\) with order type of the integers and each c.e. set in these sQ-degrees is a hypersimple set. Also, we prove that there exist two c.e. sets having no least upper bound on the \(sQ_1\)-reducibility ordering. We show that the c.e. \(sQ_1\)-degrees are not dense and if a is a c.e. \(sQ_1\)-degree such that \(o_{sQ_1}<_{sQ_1}a<_{sQ_1}o'_{sQ_1}\), then there exist infinitely many pairwise sQ-incomputable c.e. sQ-degrees \(\{c_i\}_{i\in \omega }\) such that \((\forall \,i)\;(a<_{sQ_1}c_i<_{sQ_1}o'_{sQ_1})\).