{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T06:35:12Z","timestamp":1649140512940},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":6310,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1996,12]]},"abstract":"<jats:p>This paper was inspired by Lerman [15] in which he proved various properties of upper bounds for the arithmetical degrees. We discuss the complementation property of upper bounds for the arithmetical degrees. In Lerman [15], it is proved that uniform upper bounds for the arithmetical degrees are jumps of upper bounds for the arithmetical degrees. So any uniform upper bound for the arithmetical degrees is not a minimal upper bound for the arithmetical degrees. Given a uniform upper bound <jats:italic>a<\/jats:italic> for the arithmetical degrees, we prove a minimal complementation theorem for the upper bounds for the arithmetical degrees below <jats:italic>a<\/jats:italic>. Namely, given such <jats:italic>a<\/jats:italic> and <jats:italic>b<\/jats:italic> &lt; <jats:italic>a<\/jats:italic> which is an upper bound for the arithmetical degrees, there is a minimal upper bound for the arithmetical degrees <jats:italic>c<\/jats:italic> such that <jats:italic>b<\/jats:italic> \u222a <jats:italic>c<\/jats:italic> = <jats:italic>a<\/jats:italic>. This answers a question in Lerman [15]. We prove this theorem by different methods depending on whether <jats:italic>a<\/jats:italic> has a function which is not dominated by any arithmetical function. We prove two propositions (see \u00a71), of which the theorem is an immediate consequence.<\/jats:p><jats:p>Our notation is almost standard. Let <jats:italic>A<\/jats:italic> \u2295 <jats:italic>B<\/jats:italic> = {2<jats:italic>n<\/jats:italic>\u2223<jats:italic>n<\/jats:italic> \u2208 <jats:italic>A<\/jats:italic>} \u222a {2<jats:italic>n<\/jats:italic> + 1\u2223<jats:italic>n<\/jats:italic> + 1\u2223<jats:italic>n<\/jats:italic> \u2208 <jats:italic>B<\/jats:italic>} for any sets <jats:italic>A<\/jats:italic> and <jats:italic>B<\/jats:italic>. Let <jats:italic>\u03c9<\/jats:italic> be the set of nonnegative natural numbers.<\/jats:p>","DOI":"10.2307\/2275810","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:59:54Z","timestamp":1146956394000},"page":"1158-1192","source":"Crossref","is-referenced-by-count":0,"title":["Minimal complementation below uniform upper bounds for the arithmetical degrees"],"prefix":"10.1017","volume":"61","author":[{"given":"Masahiro","family":"Kumabe","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200016832_ref002","first-page":"527","volume":"54","author":"Cooper","year":"1989","journal-title":"The strong anticupping property for recursively enumerable degrees"},{"key":"S0022481200016832_ref007","first-page":"715","volume":"43","author":"Jockusch","year":"1978","journal-title":"Double jumps of minimal degrees"},{"key":"S0022481200016832_ref004","first-page":"601","volume":"43","author":"Hodes","year":"1978","journal-title":"Uniform upper bounds on ideals of Turing degrees"},{"key":"S0022481200016832_ref009","first-page":"425","volume":"49","author":"Knight","year":"1984","journal-title":"Two theorems on degrees of models of true arithmetic"},{"key":"S0022481200016832_ref017","first-page":"714","volume":"46","author":"Posner","year":"1981","journal-title":"Degrees joining to 0\u2032"},{"key":"S0022481200016832_ref011","first-page":"434","volume":"31","author":"Lachlan","year":"1966","journal-title":"The impossibility of finding relative complements for recursively enumerable degrees"},{"key":"S0022481200016832_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90023-1"},{"key":"S0022481200016832_ref010","first-page":"516","volume":"59","author":"Kumabe","year":"1994","journal-title":"Minimal upper bounds for arithmetical degrees"},{"key":"S0022481200016832_ref016","first-page":"705","volume":"46","author":"Posner","year":"1981","journal-title":"The upper semilattice of degrees \u2264 0\u2032 is complemented"},{"key":"S0022481200016832_ref018","first-page":"714","volume-title":"Axiomatic settheory, Proc. sym. pure mat. 13 part I","author":"Sacks","year":"1971"},{"key":"S0022481200016832_ref001","unstructured":"Cooper S. B. , The jump is definable in the structure of the degrees of unsolvability, to appear, 1989."},{"key":"S0022481200016832_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1982-0660612-6"},{"key":"S0022481200016832_ref015","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(85)90001-6"},{"key":"S0022481200016832_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0090945"},{"key":"S0022481200016832_ref012","first-page":"537","volume-title":"Proceedings of the London Mathematical Society","volume":"16","author":"Lachlan","year":"1966"},{"key":"S0022481200016832_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21755-9"},{"key":"S0022481200016832_ref020","first-page":"160","volume":"54","author":"Slaman","year":"1989","journal-title":"Complementation in the Turing degrees"},{"key":"S0022481200016832_ref003","first-page":"429","volume":"35","author":"Enderton","year":"1970","journal-title":"A note on the hyperarithmetical hierarchy"},{"key":"S0022481200016832_ref019","unstructured":"Seetapun D. and Slaman T. , Minimal complement, to appear."},{"key":"S0022481200016832_ref021","first-page":"159","volume":"31","author":"Yates","year":"1966","journal-title":"A minimal pair of recursively enumerable degrees"},{"key":"S0022481200016832_ref006","first-page":"110","volume-title":"Proceedings of London Mathematical Society lecture note series","volume":"45","author":"Jockusch","year":"1980"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200016832","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,12]],"date-time":"2019-05-12T20:00:12Z","timestamp":1557691212000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200016832\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,12]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1996,12]]}},"alternative-id":["S0022481200016832"],"URL":"https:\/\/doi.org\/10.2307\/2275810","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,12]]}}}