{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,20]],"date-time":"2023-10-20T18:11:06Z","timestamp":1697825466866},"reference-count":17,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12611,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,9]]},"abstract":"<jats:p>In [6], Metakides and Nerode introduced the study of the lattice of recursively enumerable substructures of a recursively presented model as a means to understand the recursive content of certain algebraic constructions. For example, the lattice of recursively enumerable subspaces,<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048301_inline1\" \/>, of a recursively presented vector space<jats:italic>V<\/jats:italic><jats:sub>\u221e<\/jats:sub>has been studied by Kalantari, Metakides and Nerode, Retzlaff, Remmel and Shore. Similar studies have been done by Remmel [12], [13] for Boolean algebras and by Metakides and Nerode [9] for algebraically closed fields. In all of these models, the algebraic closure of a set is nontrivial. (The formal definition of the algebraic closure of a set<jats:italic>S<\/jats:italic>, denoted cl(<jats:italic>S<\/jats:italic>), is given in \u00a71, however in vector spaces, cl(<jats:italic>S<\/jats:italic>) is just the subspace generated by<jats:italic>S<\/jats:italic>, in Boolean algebras, cl(<jats:italic>S<\/jats:italic>) is just the subalgebra generated by<jats:italic>S<\/jats:italic>, and in algebraically closed fields, cl(<jats:italic>S<\/jats:italic>) is just the algebraically closed subfield generated by<jats:italic>S<\/jats:italic>.)<\/jats:p><jats:p>In this paper, we give a general model theoretic setting (whose precise definition will be given in \u00a71) in which we are able to give constructions which generalize many of the constructions of classical recursion theory. One of the main features of the models<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048301_inline2\" \/>which we study is that the algebraic closure of set<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048301_inline3\" \/>is just itself, i.e., cl(<jats:italic>S<\/jats:italic>) = S. Examples of such models include the natural numbers under equality \u3008<jats:italic>N<\/jats:italic>, = \u3009, the rational numbers under the usual ordering \u3008<jats:italic>Q<\/jats:italic>, \u2264\u3009, and a large class of<jats:italic>n<\/jats:italic>-dimensional partial orderings.<\/jats:p>","DOI":"10.2307\/2273131","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:50:56Z","timestamp":1146952256000},"page":"383-402","source":"Crossref","is-referenced-by-count":7,"title":["Recursion theory on orderings. I. A model theoretic setting"],"prefix":"10.1017","volume":"44","author":[{"given":"G.","family":"Metakides","sequence":"first","affiliation":[]},{"given":"J.B.","family":"Remmel","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048301_ref015","first-page":"162","volume":"32","author":"Robinson","year":"1967","journal-title":"Simplicity of recursively enumerable sets"},{"key":"S0022481200048301_ref014","unstructured":"Remmel J. B. , Recursion theory on orderings. II, this Journal (to appear)."},{"key":"S0022481200048301_ref013","unstructured":"Remmel J. B. , R-maximal Boolean algebras, this Journal (to appear)."},{"key":"S0022481200048301_ref012","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90027-X"},{"key":"S0022481200048301_ref011","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19550010205"},{"key":"S0022481200048301_ref010","doi-asserted-by":"publisher","DOI":"10.4064\/fm-90-1-45-52"},{"key":"S0022481200048301_ref009","unstructured":"Metakides G. , Recursion theory on fields (in preparation)."},{"key":"S0022481200048301_ref008","first-page":"A","article-title":"Creative presentations","volume":"21","author":"Metakides","year":"1947","journal-title":"Notices of the American Mathematica Society"},{"key":"S0022481200048301_ref004","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19660120125"},{"key":"S0022481200048301_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1954-0063995-6"},{"key":"S0022481200048301_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1968-0227009-1"},{"key":"S0022481200048301_ref016","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200048301_ref002","first-page":"309","volume":"23","author":"Friedberg","year":"1958","journal-title":"Three theorems on recursive enumeration, I, Decomposition, II, Maximal sets, III, Enumeration without duplication"},{"key":"S0022481200048301_ref006","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/BFb0062858","volume-title":"Algebra and Logic, Lecture Notes in Mathematics","volume":"450","author":"Metakides","year":"1975"},{"key":"S0022481200048301_ref005","unstructured":"Manaster A. B. and Remmel J. B. , \u21350-categorical dense n-dimensional partial orderings (in preparation)."},{"key":"S0022481200048301_ref017","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-65-03247-3"},{"key":"S0022481200048301_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90015-8"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048301","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,4,14]],"date-time":"2020-04-14T14:17:26Z","timestamp":1586873846000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048301\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,9]]},"references-count":17,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1979,9]]}},"alternative-id":["S0022481200048301"],"URL":"https:\/\/doi.org\/10.2307\/2273131","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,9]]}}}