{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T08:37:28Z","timestamp":1775464648511,"version":"3.50.1"},"reference-count":13,"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":17998,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1964,12]]},"abstract":"<jats:p>There are essentially two \u2018positive\u2019 results concerning the extension of effective operations to partial recursive functionals. The first proved by Myhill and Shepherdson in [9] states that any effective operation whose domain is the set of all partial recursive (p.r.) functions is potentially p.r. The second proved originally by Kreisel, Lacombe, and Shoenfield in [5] states: Let <jats:italic>F<\/jats:italic> be an effective operation mapping a set <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200115804_inline1\"\/> of recursive functions into the natural numbers; if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200115804_inline1\"\/> has a recursively dense base, then <jats:italic>F<\/jats:italic> is potentially p.r. The main objective of this paper is to present these two theorems in a general setting.<\/jats:p>","DOI":"10.2307\/2270370","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T20:26:14Z","timestamp":1146947174000},"page":"163-178","source":"Crossref","is-referenced-by-count":15,"title":["Effective operations in a general setting"],"prefix":"10.1017","volume":"29","author":[{"given":"A. H.","family":"Lachlan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200115804_ref010","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-59-02637-7"},{"key":"S0022481200115804_ref009","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19550010407"},{"key":"S0022481200115804_ref007","first-page":"129","volume-title":"Constructivity in Mathematics","author":"Lacombe","year":"1959"},{"key":"S0022481200115804_ref001","first-page":"49","article-title":"Algoritmi\u010d\u00e9ski\u00e9 operatory v konstruktivnyk polnyk s\u00e9parab\u00e9lnyk m\u00e9tri\u010d\u00e9skyk prostranstvak (Algorithmic operators on constructive complete separable metric spaces)","volume":"128","author":"\u010ceitin","year":"1959","journal-title":"Doklady Akad\u00e9mii Nauk SSSR"},{"key":"S0022481200115804_ref002","first-page":"295","article-title":"Algoritmi\u010d\u00e9ski\u00e9 operatory v konstruktivnyk m\u00e9tri\u010d\u00e9skyk prostranstvak (Algorithmic operators on constructive metric spaces)","volume":"67","author":"\u010ceitin","year":"1962","journal-title":"Trudy mat\u00e9mati\u010d\u00e9skogo instituto im\u00e8ni V. A. St\u00e9klova"},{"key":"S0022481200115804_ref006","first-page":"2478","article-title":"Extension de la notion de fonction r\u00e9cursive aux fonctions d'une ou plusieurs variables","volume":"240","author":"Lacombe","year":"1955","journal-title":"Comptes Rendus de l'Acad\u00e9mie des Sciences de Paris"},{"key":"S0022481200115804_ref004","first-page":"550","volume-title":"Introduction to Metamathematics","author":"Kleene","year":"1952"},{"key":"S0022481200115804_ref003","first-page":"378","volume":"27","author":"Gandy","year":"1962","journal-title":"Effective operations and recursive functionals, abstract"},{"key":"S0022481200115804_ref011","first-page":"528","article-title":"Generalisation of a theorem of Kreisel, Lacombe, and Shoenfield","volume":"6","author":"Pour-El","year":"1959","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200115804_ref013","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1953-0053041-6"},{"key":"S0022481200115804_ref008","unstructured":"Moschovakis J. N. , Recursive metric spaces, Ph.D. thesis, Part II, University of Wisconsin, 1963."},{"key":"S0022481200115804_ref005","first-page":"290","article-title":"Partial recursive functionals and effective operations","author":"Kreisel","journal-title":"Constructivity in Mathematics"},{"key":"S0022481200115804_ref012","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19600061510"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200115804","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,3]],"date-time":"2019-06-03T20:05:44Z","timestamp":1559592344000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200115804\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1964,12]]},"references-count":13,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1964,12]]}},"alternative-id":["S0022481200115804"],"URL":"https:\/\/doi.org\/10.2307\/2270370","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1964,12]]}}}