{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T05:46:57Z","timestamp":1772516817898,"version":"3.50.1"},"reference-count":12,"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":9142,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1989,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Posner [6] has shown, by a nonuniform proof, that every <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200027651_inline1\"\/> degree has a complement below 0\u2032. We show that a 1-generic complement for each <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200027651_inline1\"\/> set of degree between 0 and 0\u2032 can be found uniformly. Moreover, the methods just as easily can be used to produce a complement whose jump has the degree of any real recursively enumerable in and above \u2205\u2032. In the second half of the paper, we show that the complementation of the degrees below 0\u2032 does not extend to all recursively enumerable degrees. Namely, there is a pair of recursively enumerable degrees <jats:italic>a<\/jats:italic> above <jats:italic>b<\/jats:italic> such that no degree strictly below <jats:italic>a<\/jats:italic> joins <jats:italic>b<\/jats:italic> above <jats:italic>a<\/jats:italic>. (This result is independently due to S. B. Cooper.) We end with some open problems.<\/jats:p>","DOI":"10.2307\/2275022","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:29:26Z","timestamp":1146954566000},"page":"160-176","source":"Crossref","is-referenced-by-count":23,"title":["Complementation in the Turing degrees"],"prefix":"10.1017","volume":"54","author":[{"given":"Theodore A.","family":"Slaman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"John R.","family":"Steel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200027651_ref009","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-24.1.1"},{"key":"S0022481200027651_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/042\/791048"},{"key":"S0022481200027651_ref006","first-page":"705","volume":"46","author":"Posner","year":"1981","journal-title":"The upper semilattice of degrees \u0386#x2264; 0\u0386#x2032; is complemented"},{"key":"S0022481200027651_ref005","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-16.1.537"},{"key":"S0022481200027651_ref004","first-page":"434","volume":"31","author":"Lachlan","year":"1966","journal-title":"The impossibility of finding relative complements for recursively enumerable degrees"},{"key":"S0022481200027651_ref003","doi-asserted-by":"publisher","DOI":"10.2307\/1969708"},{"key":"S0022481200027651_ref001","author":"Cooper","journal-title":"The strong anticupping property for recursively enumerable degrees"},{"key":"S0022481200027651_ref007","first-page":"714","volume":"46","author":"Posner","year":"1981","journal-title":"Degrees joining to 0\u0386#x2032;"},{"key":"S0022481200027651_ref008","volume-title":"Degrees of unsolvability","author":"Sacks","year":"1966"},{"key":"S0022481200027651_ref010","unstructured":"Slaman T. A. , A recursively enumerable degree that is not the top of a diamond (to appear)."},{"key":"S0022481200027651_ref011","doi-asserted-by":"publisher","DOI":"10.2307\/1969604"},{"key":"S0022481200027651_ref012","first-page":"159","volume":"31","author":"Yates","year":"1966","journal-title":"A minimal pair of recursively 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\/S0022481200027651","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,19]],"date-time":"2019-05-19T21:01:54Z","timestamp":1558299714000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200027651\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1989,3]]},"references-count":12,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1989,3]]}},"alternative-id":["S0022481200027651"],"URL":"https:\/\/doi.org\/10.2307\/2275022","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1989,3]]}}}