{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T00:17:31Z","timestamp":1648858651209},"reference-count":5,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":5490,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1999,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p><jats:italic>K<\/jats:italic> denotes the upper semilattice of all Kleene degrees. Under ZF + AD + DC. <jats:italic>K<\/jats:italic> is well-ordered and deg(<jats:italic>X<\/jats:italic><jats:sup>SJ<\/jats:sup>) is the next Kleene degree above deg(<jats:italic>X<\/jats:italic>) for <jats:italic>X<\/jats:italic> \u2286 <jats:sup>\u03c9<\/jats:sup>\u03c9 (see [4] and [5, Chapter V]). While, without AD, properties of <jats:italic>K<\/jats:italic> are not always clear. In this note, we prove the non-distributivity of <jats:italic>K<\/jats:italic> under ZFC (\u00a71), and that of Kleene degrees between deg(<jats:italic>X<\/jats:italic>) and deg(<jats:italic>X<\/jats:italic><jats:sup>SJ<\/jats:sup>) for some <jats:italic>X<\/jats:italic> under ZFC + CH (\u00a72.3).<\/jats:p>","DOI":"10.2307\/2586756","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:02:25Z","timestamp":1146938545000},"page":"147-158","source":"Crossref","is-referenced-by-count":0,"title":["Non-distributive upper semilattice of Kleene degrees"],"prefix":"10.1017","volume":"64","author":[{"given":"Hisato","family":"Muraki","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200013955_ref003","unstructured":"Muraki H. , Non-complementedness and non-distributivity of Kleene degrees, to appear."},{"key":"S0022481200013955_ref004","first-page":"374","volume":"36","author":"Solovay","year":"1971","journal-title":"Determinacy and type 2 recursion"},{"key":"S0022481200013955_ref002","volume-title":"Constructibility","author":"Devlin"},{"key":"S0022481200013955_ref001","volume-title":"Habilitationsschrift","author":"Ambos-Spies","year":"1984"},{"key":"S0022481200013955_ref005","unstructured":"Weitbcamp G. , Kleene recursion over the continuum, Ph.D. thesis , Pennsylvania State University, 1980."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200013955","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,10]],"date-time":"2019-05-10T19:06:59Z","timestamp":1557515219000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200013955\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,3]]},"references-count":5,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1999,3]]}},"alternative-id":["S0022481200013955"],"URL":"https:\/\/doi.org\/10.2307\/2586756","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1999,3]]}}}