{"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":1772441441114,"version":"3.50.1"},"reference-count":11,"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":9050,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1989,6]]},"abstract":"<jats:p>Following Soare [11] we say a recursively enumerable (r.e.) degree <jats:bold>a<\/jats:bold> has the <jats:italic>anticupping<\/jats:italic> (a.c.) <jats:italic>property<\/jats:italic> if there is a nonzero r.e. degree <jats:bold>b<\/jats:bold> &lt; <jats:bold>a<\/jats:bold> such that for no r.e. <jats:bold>c<\/jats:bold> &lt; <jats:bold>a<\/jats:bold> does <jats:bold>a<\/jats:bold> = <jats:bold>b<\/jats:bold> \u222a <jats:bold>c<\/jats:bold>.<\/jats:p><jats:p>Cooper [2] and Yates showed that <jats:bold>0\u2032<\/jats:bold> has the a.c. property, while Harrington (see Miller [6]) proved that every high r.e. degree <jats:bold>a<\/jats:bold> has the a.c. property.<\/jats:p><jats:p>The recent paper by Ambos-Spies, Jockusch, Shore and Soare [1] describes a general theoretical framework for cupping and capping below <jats:bold>0\u2032<\/jats:bold> which seems likely to be useful in a wider context.<\/jats:p><jats:p>Definition. (1) We say <jats:bold>b<\/jats:bold> is <jats:italic>strongly noncuppable below<\/jats:italic><jats:bold>a<\/jats:bold> if <jats:bold>0 &lt; b &lt; a<\/jats:bold> and, for each <jats:bold>d &lt; a, b \u222a d \u2260 a<\/jats:bold>.<\/jats:p><jats:p>(2) We say an r.e. <jats:bold>a<\/jats:bold> has the <jats:italic>strong anticupping property<\/jats:italic> if there is an r.e. <jats:bold>b<\/jats:bold> which is strongly noncuppable below <jats:bold>a<\/jats:bold>.<\/jats:p><jats:p>The main results on cupping in <jats:bold><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002716X_inline1\"\/>(\u2264 0\u2032)<\/jats:bold> are due to Epstein, Posner and Robinson. For instance it is known (Posner and Robinson [8]) that the s.a.c. property fails for <jats:bold>0\u2032<\/jats:bold>.<\/jats:p><jats:p>We prove below that r.e. degrees with the s.a.c. property do exist, hence obtaining a nonzero r.e. degree <jats:bold>a<\/jats:bold> such that <jats:bold><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002716X_inline1\"\/>(\u2264 a) \u2262<jats:sub>e<\/jats:sub><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002716X_inline1\"\/>(\u2264h)<\/jats:bold> for any high r.e. degree <jats:bold>h<\/jats:bold>. This result, obtained by means of an infinite injury construction in <jats:bold><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002716X_inline1\"\/>(\u2264 0\u2032)<\/jats:bold>, extends Theorem 2 of [3], proved using a finite injury construction in <jats:bold><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120002716X_inline1\"\/>(\u2264 0\u2032)<\/jats:bold>.<\/jats:p><jats:p>Our main source of notation and terminology is [3].<\/jats:p>","DOI":"10.2307\/2274867","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:30:21Z","timestamp":1146940221000},"page":"527-539","source":"Crossref","is-referenced-by-count":14,"title":["The strong anticupping property for recursively enumerable degrees"],"prefix":"10.1017","volume":"54","author":[{"given":"S. B.","family":"Cooper","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120002716X_ref011","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/042\/791052"},{"key":"S002248120002716X_ref009","volume-title":"Degrees of unsoltability","volume":"55","author":"Sacks","year":"1963"},{"key":"S002248120002716X_ref007","unstructured":"Posner D. B. , High degrees, Ph.D. thesis, University of California, Berkeley, California, 1977."},{"key":"S002248120002716X_ref005","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-16.1.537"},{"key":"S002248120002716X_ref004","doi-asserted-by":"crossref","DOI":"10.1090\/memo\/0241","volume-title":"Initial segments of the degrees below 0\u2032","author":"Epstein","year":"1981"},{"key":"S002248120002716X_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(87)90039-X"},{"key":"S002248120002716X_ref002","unstructured":"Cooper S. B. , On a theorem of C. E. M. Yates, handwritten notes, 1974."},{"key":"S002248120002716X_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1984-0719661-8"},{"key":"S002248120002716X_ref008","first-page":"714","volume":"46","author":"Posner","year":"1981","journal-title":"Degrees joining to 0\u2032"},{"key":"S002248120002716X_ref010","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-24.1.1"},{"key":"S002248120002716X_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0090950"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120002716X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,19]],"date-time":"2019-05-19T16:26:25Z","timestamp":1558283185000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120002716X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1989,6]]},"references-count":11,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1989,6]]}},"alternative-id":["S002248120002716X"],"URL":"https:\/\/doi.org\/10.2307\/2274867","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1989,6]]}}}