{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T12:04:39Z","timestamp":1759147479054},"reference-count":18,"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":2110,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2008,6]]},"abstract":"<jats:p><jats:bold>\u00a71. Introduction<\/jats:bold>. Sacks [16] showed that every computably enumerable (c.e.) degree &gt; <jats:italic>0<\/jats:italic> has a c.e. splitting. Hence, relativising, every c.e. degree has a \u0394<jats:sub>2<\/jats:sub> splitting above each proper predecessor (by \u2018splitting\u2019 we understand \u2018nontrivial splitting\u2019). Arslanov [1] showed that <jats:italic>0<\/jats:italic>\u2032 has a d.c.e. splitting above each c.e. <jats:italic>a<\/jats:italic> &lt; <jats:italic>0<\/jats:italic>\u2032. On the other hand, Lachlan [11] proved the existence of a c.e. <jats:italic>a<\/jats:italic> &lt; <jats:italic>0<\/jats:italic> which has no c.e. splitting above some proper c.e. predecessor, and Harrington [10] showed that one could take <jats:italic>a<\/jats:italic> = <jats:italic>0<\/jats:italic>\u2032. Splitting and nonsplitting techniques have had a number of consequences for definability and elementary equivalence in the degrees below <jats:italic>0<\/jats:italic>\u2032.<\/jats:p><jats:p>Heterogeneous splittings are best considered in the context of cupping and non-cupping. Posner and Robinson [15] showed that every nonzero \u0394<jats:sub>2<\/jats:sub> degree can be nontrivially cupped to <jats:italic>0<\/jats:italic>\u2032, and Arslanov [1] showed that every c.e. degree &gt; <jats:italic>0<\/jats:italic> can be d.c.e. cupped to <jats:italic>0<\/jats:italic>\u2032 (and hence since every d.c.e., or even n-c.e., degree has a nonzero c.e. predecessor, every n-c.e. degree &gt; <jats:italic>0<\/jats:italic> is d.c.e. cuppable). Cooper [4] and Yates (see Miller [13]) showed the existence of degrees noncuppable in the c.e. degrees. Moreover, the search for relative cupping results was drastically limited by Cooper [5], and Slaman and Steel [17] (see also Downey [9]), who showed that there is a nonzero c.e. degree <jats:italic>a<\/jats:italic> below which even \u0394<jats:sub>2<\/jats:sub> cupping of c.e. degrees fails.<\/jats:p><jats:p>We prove below what appears to be the strongest possible of such nonsplitting and noncupping results.<\/jats:p>","DOI":"10.2178\/jsl\/1208359064","type":"journal-article","created":{"date-parts":[[2008,4,16]],"date-time":"2008-04-16T19:50:47Z","timestamp":1208375447000},"page":"634-655","source":"Crossref","is-referenced-by-count":7,"title":["How enumeration reductibility yields extended Harrington non-splitting"],"prefix":"10.1017","volume":"73","author":[{"given":"Mariya I.","family":"Soskova","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S. Barry","family":"Cooper","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200004540_ref018","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02460-7"},{"key":"S0022481200004540_ref017","first-page":"160","volume":"54","author":"Slaman","year":"1989","journal-title":"Complementation in the Turing degrees"},{"key":"S0022481200004540_ref016","doi-asserted-by":"publisher","DOI":"10.2307\/1970214"},{"key":"S0022481200004540_ref014","volume-title":"Classical Recursion Theory, Volume II","author":"Odifreddi","year":"1999"},{"key":"S0022481200004540_ref010","unstructured":"Harrington L. , Understanding Lachlan s monster paper, 1980, handwritten notes."},{"key":"S0022481200004540_ref009","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1215\/ijm\/1256069291","article-title":"degrees and transfer theorems","volume":"31","author":"Downey","year":"1987","journal-title":"Illinois Journal of Mathematics"},{"key":"S0022481200004540_ref007","volume-title":"Computability Theory","author":"Cooper","year":"2004"},{"key":"S0022481200004540_ref004","unstructured":"Cooper S. B. , On a theorem of C. E. M. Yates, 1974, handwritten notes."},{"key":"S0022481200004540_ref002","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022660222520"},{"key":"S0022481200004540_ref001","first-page":"270","article-title":"Structural properties of the degrees below 0\u2032","volume":"283","author":"Arslanov","year":"1985","journal-title":"Doklady Akademii Nauk SSSR"},{"key":"S0022481200004540_ref003","volume-title":"Logic Colloquium'98","author":"Arslanov","year":"1999"},{"key":"S0022481200004540_ref015","first-page":"714","volume":"46","author":"Posner","year":"1981","journal-title":"Degrees joining to 0\u2032"},{"key":"S0022481200004540_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/BF01794984"},{"key":"S0022481200004540_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0090950"},{"key":"S0022481200004540_ref008","first-page":"741","volume":"70","author":"Cooper","year":"2005","journal-title":"Bounding and nonbounding minimal pairs in the enumeration degrees"},{"key":"S0022481200004540_ref006","first-page":"57","volume-title":"Recursion Theory Week, Proceedings Oberwolfach 1989","volume":"1432","author":"Cooper","year":"1990"},{"key":"S0022481200004540_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90016-4"},{"key":"S0022481200004540_ref005","first-page":"527","volume":"54","author":"Cooper","year":"1989","journal-title":"The strong anti-cupping property for 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\/S0022481200004540","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,30]],"date-time":"2019-04-30T23:32:40Z","timestamp":1556667160000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200004540\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,6]]},"references-count":18,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2008,6]]}},"alternative-id":["S0022481200004540"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1208359064","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,6]]}}}