7 found
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  1.  12
    Strong Enumeration Reducibilities.Roland Sh Omanadze & Andrea Sorbi - 2006 - Archive for Mathematical Logic 45 (7):869-912.
    We investigate strong versions of enumeration reducibility, the most important one being s-reducibility. We prove that every countable distributive lattice is embeddable into the local structure $L(\mathfrak D_s)$ of the s-degrees. However, $L(\mathfrak D_s)$ is not distributive. We show that on $\Delta^{0}_{2}$ sets s-reducibility coincides with its finite branch version; the same holds of e-reducibility. We prove some density results for $L(\mathfrak D_s)$ . In particular $L(\mathfrak D_s)$ is upwards dense. Among the results about reducibilities that are stronger than s-reducibility, (...)
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  2.  9
    Some structural properties of quasi-degrees.Roland Sh Omanadze - 2018 - Logic Journal of the IGPL 26 (1):191-201.
  3.  12
    A characterization of the Δ⁰₂ hyperhyperimmune sets.Roland Sh Omanadze & Andrea Sorbi - 2008 - Journal of Symbolic Logic 73 (4):1407-1415.
    Let A be an infinite Δ₂⁰ set and let K be creative: we show that K≤Q A if and only if K≤Q₁ A. (Here ≤Q denotes Q-reducibility, and ≤Q₁ is the subreducibility of ≤Q obtained by requesting that Q-reducibility be provided by a computable function f such that Wf(x)∩ Wf(y)=∅, if x \not= y.) Using this result we prove that A is hyperhyperimmune if and only if no Δ⁰₂ subset B of A is s-complete, i.e., there is no Δ⁰₂ subset (...)
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  4.  18
    Immunity properties and strong positive reducibilities.Irakli O. Chitaia, Roland Sh Omanadze & Andrea Sorbi - 2011 - Archive for Mathematical Logic 50 (3-4):341-352.
    We use certain strong Q-reducibilities, and their corresponding strong positive reducibilities, to characterize the hyperimmune sets and the hyperhyperimmune sets: if A is any infinite set then A is hyperimmune (respectively, hyperhyperimmune) if and only if for every infinite subset B of A, one has \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{K}\not\le_{\rm ss} B}$$\end{document} (respectively, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{K}\not\le_{\overline{\rm s}} B}$$\end{document}): here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\le_{\overline{\rm (...)
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  5.  10
    On the bounded quasi‐degrees of c.e. sets.Roland Sh Omanadze - 2013 - Mathematical Logic Quarterly 59 (3):238-246.
  6.  2
    r‐Maximal sets and Q1,N‐reducibility.Roland Sh Omanadze & Irakli O. Chitaia - 2021 - Mathematical Logic Quarterly 67 (2):138-148.
    We show that if M is an r‐maximal set, A is a major subset of M, B is an arbitrary set and, then. We prove that the c.e. ‐degrees are not dense. We also show that there exist infinite collections of ‐degrees and such that the following hold: (i) for every i, j,, and,(ii) each consists entirely of r‐maximal sets, and(iii) each consists entirely of non‐r‐maximal hyperhypersimple sets.
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  7.  2
    $$sQ_1$$ -degrees of computably enumerable sets.Roland Sh Omanadze - 2023 - Archive for Mathematical Logic 62 (3):401-417.
    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 (...)
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