{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,18]],"date-time":"2025-05-18T05:22:05Z","timestamp":1747545725779},"reference-count":9,"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":8412,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1991,3]]},"abstract":"<jats:p>Lachlan [5] has shown that it is not possible to embed the diamond lattice in the r.e. Turing degrees while preserving least and greatest elements; that is, there do not exist incomparable r.e. Turing degrees <jats:bold>a<\/jats:bold> and <jats:bold>b<\/jats:bold> such that <jats:bold>a<\/jats:bold> \u2227 <jats:bold>b<\/jats:bold> = <jats:bold>0<\/jats:bold> and <jats:bold>a<\/jats:bold> \u2228 <jats:bold>b<\/jats:bold> = <jats:bold>0\u2032<\/jats:bold>. Cooper [3] has compared the r.e. Turing degrees to the enumeration degrees below <jats:bold>0<\/jats:bold><jats:sub><jats:italic>e<\/jats:italic><\/jats:sub>\u2032 and has asked if the two structures are elementarily equivalent.<\/jats:p><jats:p>In this paper we show that such an embedding is possible in the <jats:italic>\u03a3<\/jats:italic><jats:sub>2<\/jats:sub><jats:italic>enumeration degrees<\/jats:italic>, which implies a negative answer to Cooper's question.<\/jats:p><jats:p>Theorem. <jats:italic>There are low enumeration degrees<\/jats:italic><jats:bold>a<\/jats:bold><jats:italic>and<\/jats:italic><jats:bold>b<\/jats:bold><jats:italic>such that<\/jats:italic><jats:bold>a<\/jats:bold> \u2227 <jats:bold>b<\/jats:bold> = <jats:bold>0<\/jats:bold><jats:sub>e<\/jats:sub><jats:italic>and<\/jats:italic><jats:bold>a<\/jats:bold> \u2228 <jats:bold>b<\/jats:bold> = <jats:bold>0<\/jats:bold><jats:sub>e<\/jats:sub>\u2032.<\/jats:p><jats:p>Lower case italic letters denote elements of <jats:italic>\u03c9<\/jats:italic> while upper case italic letters denote subsets of <jats:italic>\u03c9<\/jats:italic>. <jats:italic>D, E<\/jats:italic> and <jats:italic>F<\/jats:italic> are reserved for finite sets, and <jats:italic>K<\/jats:italic> for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025007_inline2\" \/>\u2032. If <jats:italic>D<\/jats:italic> = {<jats:italic>x<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>} then the <jats:italic>canonical index<\/jats:italic> of <jats:italic>D<\/jats:italic> is <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025007_inline1\" \/>, and the canonical index of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025007_inline2\" \/> is \u2205. <jats:italic>D<jats:sub>x<\/jats:sub><\/jats:italic> denotes the set with canonical index <jats:italic>x<\/jats:italic>. {<jats:italic>W<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>}<jats:sub><jats:italic>i\u2208\u03c9<\/jats:italic><\/jats:sub> is any fixed standard listing of the r.e. sets, and &lt;\u00b7, \u00b7&gt; is any fixed recursive bijection from <jats:italic>\u03c9<\/jats:italic> \u00d7 <jats:italic>\u03c9<\/jats:italic> to <jats:italic>\u03c9<\/jats:italic>.<\/jats:p><jats:p>Intuitively, <jats:italic>A<\/jats:italic> is enumeration reducible to <jats:italic>B<\/jats:italic> if there is an effective algorithm for producing an enumeration of <jats:italic>A<\/jats:italic> from <jats:italic>any<\/jats:italic> enumeration of <jats:italic>B<\/jats:italic>. There is a natural one-to-one correspondence between all such algorithms and the r.e. sets.<\/jats:p>","DOI":"10.2307\/2274914","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:38:25Z","timestamp":1146955105000},"page":"195-212","source":"Crossref","is-referenced-by-count":13,"title":["Embedding the diamond in the \u03a3<sub>2<\/sub> enumeration degrees"],"prefix":"10.1017","volume":"56","author":[{"given":"Seema","family":"Ahmad","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200025007_ref008","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200025007_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90011-3"},{"key":"S0022481200025007_ref003","first-page":"503","volume":"49","author":"Cooper","year":"1984","journal-title":"Partial degrees and the density problem. Part 2: The enumeration degrees of the \u03a32 sets are dense"},{"key":"S0022481200025007_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02460-7"},{"key":"S0022481200025007_ref002","first-page":"854","volume":"47","author":"Cooper","year":"1982","journal-title":"Partial degrees and the density problem"},{"key":"S0022481200025007_ref004","unstructured":"Gutteridge L. , Some results on enumeration reducibility, Ph.D. Dissertation, Simon Fraser University, Burnaby, 1971."},{"key":"S0022481200025007_ref005","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-16.1.537"},{"key":"S0022481200025007_ref006","first-page":"839","volume":"50","author":"McEvoy","year":"1985","journal-title":"Jumps of quasi-minimal enumeration degrees"},{"key":"S0022481200025007_ref007","first-page":"983","volume":"50","author":"McEvoy","year":"1985","journal-title":"On minimal pairs of enumeration degrees"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200025007","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T22:36:47Z","timestamp":1558132607000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200025007\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1991,3]]},"references-count":9,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1991,3]]}},"alternative-id":["S0022481200025007"],"URL":"https:\/\/doi.org\/10.2307\/2274914","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1991,3]]}}}