{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,2,1]],"date-time":"2024-02-01T23:26:07Z","timestamp":1706829967584},"reference-count":28,"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":12429,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1980,3]]},"abstract":"<jats:p>A natural way of studying the computability of an algebraic structure or process is to apply some of the theory of the recursive functions to the algebra under consideration through the manufacture of appropriate coordinate systems from the natural numbers. An algebraic structure<jats:italic>A<\/jats:italic>= (<jats:italic>A<\/jats:italic>;<jats:italic>\u03c3<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026,<jats:italic>\u03c3<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>) is<jats:italic>computable<\/jats:italic>if it possesses a recursive coordinate system in the following precise sense: associated to<jats:italic>A<\/jats:italic>there is a pair (<jats:italic>\u03b1, \u03a9<\/jats:italic>) consisting of a recursive set of natural numbers<jats:italic>\u03a9<\/jats:italic>and a surjection<jats:italic>\u03b1<\/jats:italic>:<jats:italic>\u03a9<\/jats:italic>\u2192<jats:italic>A<\/jats:italic>so that (i) the relation defined on<jats:italic>\u03a9<\/jats:italic>by<jats:italic>n<\/jats:italic>\u2261<jats:italic><jats:sub>\u03b1<\/jats:sub>m<\/jats:italic>iff<jats:italic>\u03b1<\/jats:italic>(<jats:italic>n<\/jats:italic>) =<jats:italic>\u03b1<\/jats:italic>(<jats:italic>m<\/jats:italic>) in<jats:italic>A<\/jats:italic>is recursive, and (ii) each of the operations of<jats:italic>A<\/jats:italic>may be effectively followed in<jats:italic>\u03a9<\/jats:italic>, that is, for each (say)<jats:italic>r<\/jats:italic>-ary operation<jats:italic>\u03c3<\/jats:italic>on<jats:italic>A<\/jats:italic>there is an<jats:italic>r<\/jats:italic>argument recursive function<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047198_inline1\" \/>on<jats:italic>\u03a9<\/jats:italic>which commutes the diagram<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200047198_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>wherein<jats:italic>\u03b1<jats:sup>r<\/jats:sup><\/jats:italic>is<jats:italic>r<\/jats:italic>-fold<jats:italic>\u03b1<\/jats:italic>\u00d7 \u2026 \u00d7<jats:italic>\u03b1<\/jats:italic>.<\/jats:p><jats:p>This concept of a computable algebraic system is the independent technical idea of M.O.Rabin [18] and A.I.Mal'cev [14]. From these first papers one may learn of the strength and elegance of the general method of coordinatising; note-worthy for us is the fact that computability is a finiteness condition of algebra\u2014an isomorphism invariant possessed of all finite algebraic systems\u2014and that it serves to set upon an algebraic foundation the combinatorial idea that a system can be combinatorially presented and have effectively decidable term or word problem.<\/jats:p>","DOI":"10.2307\/2273358","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:51:56Z","timestamp":1146952316000},"page":"103-120","source":"Crossref","is-referenced-by-count":7,"title":["Computability and the algebra of fields: Some affine constructions"],"prefix":"10.1017","volume":"45","author":[{"given":"J. V.","family":"Tucker","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200047198_ref028","volume-title":"Algebra. I","author":"Waerden","year":"1970"},{"key":"S0022481200047198_ref027","volume-title":"Modern algebra. I","author":"Waerden","year":"1949"},{"key":"S0022481200047198_ref025","unstructured":"Tucker J.V. , Computability as an algebraic property. Part one: general theory (in preparation)."},{"key":"S0022481200047198_ref023","doi-asserted-by":"crossref","DOI":"10.1525\/9780520348097","volume-title":"A decision method for elementary algebra and geometry","author":"Tarski","year":"1951"},{"key":"S0022481200047198_ref021","volume-title":"An introduction to non-associative algebras","author":"Schafer","year":"1966"},{"key":"S0022481200047198_ref016","volume-title":"On group-theoretic decision problems and their ctassification","author":"Miller","year":"1971"},{"key":"S0022481200047198_ref013","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-4.2.304"},{"key":"S0022481200047198_ref011","first-page":"1","volume-title":"Proceedings of the Liverpool Singularities Symposium 1","author":"Levine","year":"1971"},{"key":"S0022481200047198_ref009","volume-title":"Elements of mathematical logic","author":"Kreisel","year":"1971"},{"key":"S0022481200047198_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9443-3"},{"key":"S0022481200047198_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71185-6"},{"key":"S0022481200047198_ref005","doi-asserted-by":"publisher","DOI":"10.1137\/1011034"},{"key":"S0022481200047198_ref004","volume-title":"Universal algebra","author":"Cohn","year":"1965"},{"key":"S0022481200047198_ref002","volume-title":"Selected papers on mathematical trends in control theory","author":"Bellman","year":"1964"},{"key":"S0022481200047198_ref001","volume-title":"Ordinary differential equations","author":"Arnold","year":"1973"},{"key":"S0022481200047198_ref017","first-page":"40","article-title":"Notation systems and recursive ordered fields","volume":"17","author":"Moschovakis","year":"1965","journal-title":"Compositio Mathematica"},{"key":"S0022481200047198_ref022","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-96200-4"},{"key":"S0022481200047198_ref007","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.1956.0003"},{"key":"S0022481200047198_ref024","first-page":"164","volume-title":"The philosophy of mathematics","author":"Tarski","year":"1969"},{"key":"S0022481200047198_ref018","first-page":"341","article-title":"Computable algebra, general theory and the theory of computable fields","volume":"95","author":"Rabin","year":"1960","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200047198_ref019","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1954-0063328-5"},{"key":"S0022481200047198_ref003","volume-title":"Mathematical developments arising from Hubert's problems","author":"Browder","year":"1976"},{"key":"S0022481200047198_ref012","first-page":"239","volume":"35","author":"Madison","year":"1970","journal-title":"A note on computable real fields"},{"key":"S0022481200047198_ref014","first-page":"148","volume-title":"The meta-mathematics of algebraic systems. Collected papers: 1967\u20131976","author":"Mal'cev","year":"1971"},{"key":"S0022481200047198_ref026","unstructured":"Tucker J.V. , Computability as an algebraic property. Part two: applications (in preparation)."},{"key":"S0022481200047198_ref020","doi-asserted-by":"publisher","DOI":"10.2307\/1969640"},{"key":"S0022481200047198_ref015","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-65374-2"},{"key":"S0022481200047198_ref010","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1970-0253897-3"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200047198","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,26]],"date-time":"2021-07-26T20:20:12Z","timestamp":1627330812000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200047198\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1980,3]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1980,3]]}},"alternative-id":["S0022481200047198"],"URL":"https:\/\/doi.org\/10.2307\/2273358","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1980,3]]}}}