Muraki, Hisato Non-distributive upper semilattice of Kleene degrees. (English) Zbl 0926.03048 J. Symb. Log. 64, No. 1, 147-158 (1999). Summary: \({\mathcal K}\) denotes the upper semilattice of all Kleene degrees. Under ZF+AD+DC, \({\mathcal K}\) is well-ordered and \(\deg(X^{SJ})\) is the next Kleene degree above \(\deg(X)\) for \(X\subseteq{^\omega \omega}\). While, without AD, properties of \({\mathcal K}\) are not always clear. In this note, we prove the non-distributivity of \({\mathcal K}\) under ZFC, and that of Kleene degrees between \(\deg(X)\) and \(\deg(X^{SJ})\) for some \(X\) under ZFC+CH. MSC: 03D65 Higher-type and set recursion theory 03D30 Other degrees and reducibilities in computability and recursion theory 03E60 Determinacy principles Keywords:axiom of determinacy; upper semilattice of Kleene degrees; Kleene recursive; Kleene semirecursive; non-distributivity PDFBibTeX XMLCite \textit{H. Muraki}, J. Symb. Log. 64, No. 1, 147--158 (1999; Zbl 0926.03048) Full Text: DOI References: [1] Determinacy and type 2 recursion 36 pp 374– (1971) [2] Habilitationsschrift (1984) [3] Constructibility · Zbl 0542.03029 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.