$$sQ_1$$ -degrees of computably enumerable sets

Archive for Mathematical Logic 62 (3):401-417 (2023)
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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}, then there exist infinitely many pairwise _sQ_-incomputable c.e. _sQ_-degrees \(\{c_i\}_{i\in \omega }\) such that \((\forall \,i)\;(a.

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References found in this work

Degree structures of conjunctive reducibility.Irakli Chitaia & Roland Omanadze - 2021 - Archive for Mathematical Logic 61 (1):19-31.
Hyperhypersimple sets and Q1 -reducibility.Irakli Chitaia - 2016 - Mathematical Logic Quarterly 62 (6):590-595.
Q1-degrees of c.e. sets.R. Sh Omanadze & Irakli O. Chitaia - 2012 - Archive for Mathematical Logic 51 (5-6):503-515.
Strong Enumeration Reducibilities.Roland Sh Omanadze & Andrea Sorbi - 2006 - Archive for Mathematical Logic 45 (7):869-912.
Computably enumerable sets and quasi-reducibility.R. Downey, G. LaForte & A. Nies - 1998 - Annals of Pure and Applied Logic 95 (1-3):1-35.

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