{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T19:42:31Z","timestamp":1649101351568},"reference-count":3,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":4119,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,12]]},"abstract":"<jats:p>A computably enumerable Turing degree <jats:bold><jats:italic>a<\/jats:italic><\/jats:bold> is called <jats:italic>contiguous<\/jats:italic> iff it contains only a single computably enumerable weak truth table degree (Ladner and Sasso [2]). In [1], the authors proved that a nonzero computably enumerable degree <jats:bold><jats:italic>a<\/jats:italic><\/jats:bold> is contiguous iff it is <jats:italic>locally distributive<\/jats:italic>, that is, for all <jats:bold><jats:italic>a<\/jats:italic><\/jats:bold><jats:sub>1<\/jats:sub>, <jats:bold><jats:italic>a<\/jats:italic><\/jats:bold><jats:sub>2<\/jats:sub>, <jats:bold><jats:italic>c<\/jats:italic><\/jats:bold> with <jats:bold><jats:italic>a<\/jats:italic><\/jats:bold><jats:sub>1<\/jats:sub> \u222a<jats:bold><jats:italic>a<\/jats:italic><\/jats:bold><jats:sub>2<\/jats:sub> = <jats:bold><jats:italic>a<\/jats:italic><\/jats:bold> and <jats:bold><jats:italic>c<\/jats:italic><\/jats:bold> \u2264 <jats:bold><jats:italic>a<\/jats:italic><\/jats:bold>, there exist <jats:bold><jats:italic>c<\/jats:italic><\/jats:bold><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>, \u2264 <jats:bold><jats:italic>a<\/jats:italic><\/jats:bold><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> with <jats:bold><jats:italic>c<\/jats:italic><\/jats:bold><jats:sub>1<\/jats:sub> \u222a <jats:bold><jats:italic>c<\/jats:italic><\/jats:bold><jats:sub>2<\/jats:sub> = <jats:bold><jats:italic>c<\/jats:italic><\/jats:bold>.<\/jats:p><jats:p>To do this we supposed that <jats:italic>W<\/jats:italic> was a computably enumerable set and \u222a a computably set with a Turing functional \u03a6 such that \u03a6<jats:sup><jats:italic>W<\/jats:italic><\/jats:sup> = <jats:italic>U<\/jats:italic>. Then we constructed computably enumerable sets <jats:italic>A<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>A<\/jats:italic><jats:sub>1<\/jats:sub> and <jats:italic>B<\/jats:italic> together with functionals \u0393<jats:sub>0<\/jats:sub>, \u0393<jats:sub>1<\/jats:sub>, \u0393, and \u0394 so that<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200009221_Uequ1\" \/><\/jats:disp-formula><\/jats:p><jats:p>and so as to satisfy all the requirements below.<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200009221_Uequ2\" \/><\/jats:disp-formula><\/jats:p><jats:p>That is, we built a degree-theoretical splitting <jats:italic>A<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>A<\/jats:italic><jats:sub>1<\/jats:sub> of <jats:italic>W<\/jats:italic> and a set <jats:italic>B<\/jats:italic> \u2264<jats:sub><jats:italic>T<\/jats:italic><\/jats:sub><jats:italic>W<\/jats:italic> such that if we cannot beat all possible degree-theoretical splittings <jats:italic>V<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>V<\/jats:italic><jats:sub>1<\/jats:sub> of <jats:italic>B<\/jats:italic> then we were able to witness the fact that <jats:italic>U<\/jats:italic> \u2264<jats:sub><jats:italic>W<\/jats:italic><\/jats:sub><jats:italic>W<\/jats:italic> (via \u039b).<\/jats:p><jats:p>After the proof it was observed that the set <jats:italic>U<\/jats:italic> of the proof (page 1222, paragraph 4) needed only to be \u0394<jats:sub arrange=\"stack\">2<\/jats:sub><jats:sup arrange=\"stack\">0<\/jats:sup>. It was then claimed that a consequence to the proof was that every contiguous computably enumerable degree was, in fact, <jats:italic>strongly contiguous<\/jats:italic>, in the sense that all (not necessarily computably enumerable) sets of the degree had the same weak truth table degree.<\/jats:p>","DOI":"10.2178\/jsl\/1190150300","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T14:16:54Z","timestamp":1197555414000},"page":"1579-1580","source":"Crossref","is-referenced-by-count":0,"title":["Contiguity and distributivity in the enumerable Turing degrees \u2014 Corrigendum"],"prefix":"10.1017","volume":"67","author":[{"given":"Rodney G.","family":"Downey","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Steffen","family":"Lempp","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200009221_ref003","unstructured":"Nies A. , personal communication, 01 2001."},{"key":"S0022481200009221_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(75)90007-8"},{"key":"S0022481200009221_ref001","first-page":"1215","volume":"62","author":"Downey","year":"1997","journal-title":"Contiguity and distributivity in the enumerable degrees"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200009221","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T20:17:52Z","timestamp":1557173872000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200009221\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,12]]},"references-count":3,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2002,12]]}},"alternative-id":["S0022481200009221"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1190150300","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,12]]}}}