{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,16]],"date-time":"2026-02-16T08:45:51Z","timestamp":1771231551056,"version":"3.50.1"},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":1745,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2009,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Khutoretskii's Theorem states that the Rogers semilattice of any family of c.e. sets has either at most one or infinitely many elements. A lemma in the inductive step of the proof shows that no Rogers semilattice can be partitioned into a principal ideal and a principal filter. We show that such a partitioning is possible for some family of d.c.e. sets. In fact, we construct a family of c.e. sets which, when viewed as a family of d.c.e. sets, has (up to equivalence) exactly two computable Friedberg numberings <jats:italic>\u03bc<\/jats:italic> and <jats:italic>\u03bd<\/jats:italic>, and <jats:italic>\u03bc<\/jats:italic> reduces to any computable numbering not equivalent to <jats:italic>\u03bd<\/jats:italic>. The question of whether the full statement of Khutoretskii's Theorem fails for families of d.c.e. sets remains open.<\/jats:p>","DOI":"10.2178\/jsl\/1243948330","type":"journal-article","created":{"date-parts":[[2009,6,2]],"date-time":"2009-06-02T09:12:25Z","timestamp":1243933945000},"page":"618-640","source":"Crossref","is-referenced-by-count":20,"title":["A decomposition of the Rogers semilattice of a family of d.c.e. sets"],"prefix":"10.1017","volume":"74","author":[{"given":"Serikzhan A.","family":"Badaev","sequence":"first","affiliation":[]},{"given":"Steffen","family":"Lempp","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200003595_ref016","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02460-7"},{"key":"S0022481200003595_ref015","unstructured":"[LeLN] Lempp Steffen , Lecture notes on priority arguments, preprint available at http:\/\/www.math.wisc.edu\/~lempp\/papers\/prio.pdf."},{"key":"S0022481200003595_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/BF02219842"},{"key":"S0022481200003595_ref013","first-page":"52","article-title":"A unique positive enumeration","volume":"4","author":"Goncharov","year":"1994","journal-title":"Siberian Advances in Mathematics"},{"key":"S0022481200003595_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/BF01982111"},{"key":"S0022481200003595_ref008","volume-title":"Theory of numerations","author":"Ershov","year":"1977"},{"key":"S0022481200003595_ref006","first-page":"473","article-title":"Theory of numerations. Part IT. Computable numerations of morphisms","volume":"21","author":"Ershov","year":"1975","journal-title":"Zeitschrift f\u00fcr Mathematische Logik und Grundlagen der Mathematik"},{"key":"S0022481200003595_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-0755-0_4"},{"key":"S0022481200003595_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-0755-0_2"},{"key":"S0022481200003595_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/257\/04025"},{"key":"S0022481200003595_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-0755-0_3"},{"key":"S0022481200003595_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(99)80030-5"},{"key":"S0022481200003595_ref011","first-page":"4","volume-title":"Trudy Inst. Matem. SO AN SSSR","volume":"2","author":"Goncharov","year":"1982"},{"key":"S0022481200003595_ref005","first-page":"289","article-title":"Theory of numerations. 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Part III: Constructive models","volume":"23","author":"Ershov","year":"1977","journal-title":"Zeitschrift f\u00fcr Mathematische Logik und Grundlagen der Mathematik"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200003595","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,30]],"date-time":"2019-04-30T09:26:54Z","timestamp":1556616414000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200003595\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,6]]},"references-count":16,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2009,6]]}},"alternative-id":["S0022481200003595"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1243948330","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,6]]}}}