{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T08:50:41Z","timestamp":1772441441380,"version":"3.50.1"},"reference-count":0,"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":13798,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1976,6]]},"abstract":"<jats:p>In this paper we show that any countable upper semilattice with zero can be embedded as an initial segment of the degrees of unsolvability. This provides a characterization of the order types of the countable initial segments of the degrees since any such initial segment is trivially an initial segment of a countable upper semilattice. Let a segment <jats:italic>S<\/jats:italic> of the degrees be a set of degrees such that, if <jats:italic>a, b<\/jats:italic> \u0404 <jats:italic>S<\/jats:italic> and <jats:italic>a<\/jats:italic> &lt; <jats:italic>c<\/jats:italic> &lt; <jats:italic>b<\/jats:italic>, then <jats:italic>c<\/jats:italic> \u0404 <jats:italic>S<\/jats:italic>. One may readily observe that our result also characterizes the order types of all countable segments of the degrees of unsolvability.<\/jats:p>","DOI":"10.2307\/2272227","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:41:13Z","timestamp":1146951673000},"page":"289-300","source":"Crossref","is-referenced-by-count":33,"title":["Countable initial segments of the degrees of unsolvability"],"prefix":"10.1017","volume":"41","author":[{"given":"A. H.","family":"Lachlan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R.","family":"Lebeuf","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200051343","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,24]],"date-time":"2023-03-24T03:09:51Z","timestamp":1679627391000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200051343\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1976,6]]},"references-count":0,"aliases":["10.1017\/s0022481200051343"],"journal-issue":{"issue":"2","published-print":{"date-parts":[[1976,6]]}},"alternative-id":["S0022481200051343"],"URL":"https:\/\/doi.org\/10.2307\/2272227","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1976,6]]}}}