{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T23:51:58Z","timestamp":1648770718473},"reference-count":10,"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":12976,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,9]]},"abstract":"<jats:p>This paper gives some simple existence results on prime and atomic models over sets. It also contains an example in which there is no prime model over a certain set even though there is an atomic model over the set. The existence results are \u201clocal\u201d in that they deal with just one set rather than all sets contained in models of some theory. For contrast, see the \u201cglobal\u201d results in [6] or [7, p. 200].<\/jats:p><jats:p>Throughout the paper, <jats:italic>L<\/jats:italic> is a countable language, and <jats:italic>T<\/jats:italic> is a complete <jats:italic>L<\/jats:italic>-theory with infinite models. There is a \u201clarge\u201d model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049288_inline1\" \/> of <jats:italic>T<\/jats:italic> that contains the set <jats:italic>X<\/jats:italic> and any other sets and models to be used in a particular construction of a prime or atomic model over <jats:italic>X<\/jats:italic>.<\/jats:p><jats:p>A model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049288_inline2\" \/> is said to be <jats:italic>prime<\/jats:italic> over <jats:italic>X<\/jats:italic> if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049288_inline3\" \/> and every elementary monomorphism on <jats:italic>X<\/jats:italic> can be extended to an elementary embedding on all of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049288_inline4\" \/>. This notion is used in a variety of ways in model theory. It aids in distinguishing between models that are not isomorphic, as in Vaught [10]. It also aids in showing that certain models <jats:italic>are<\/jats:italic> isomorphic, as in Baldwin and Lachlan [1].<\/jats:p>","DOI":"10.2307\/2273516","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:48:21Z","timestamp":1146937701000},"page":"385-393","source":"Crossref","is-referenced-by-count":2,"title":["Prime and atomic models"],"prefix":"10.1017","volume":"43","author":[{"given":"Julia F.","family":"Knight","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049288_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BF02762619"},{"key":"S0022481200049288_ref001","first-page":"79","volume":"36","author":"Baldwin","year":"1971","journal-title":"On strongly minimal sets"},{"key":"S0022481200049288_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(74)90018-7"},{"key":"S0022481200049288_ref010","first-page":"303","volume-title":"Infinitistic methods, Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 1959","author":"Vaught","year":"1961"},{"key":"S0022481200049288_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF02771568"},{"key":"S0022481200049288_ref004","doi-asserted-by":"crossref","unstructured":"Makkai M. , An admissible generalization of a theorem on countable \u03a31 1-sets with applications, preprint.","DOI":"10.1016\/0003-4843(77)90008-0"},{"key":"S0022481200049288_ref007","volume-title":"Saturated model theory","author":"Sacks","year":"1972"},{"key":"S0022481200049288_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1965-0175782-0"},{"key":"S0022481200049288_ref008","first-page":"107","volume":"37","author":"Shelah","year":"1972","journal-title":"Uniqueness and characterization of prime models over sets for totally transcendental first order theories"},{"key":"S0022481200049288_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757882"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049288","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T15:33:27Z","timestamp":1558971207000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049288\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,9]]},"references-count":10,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1978,9]]}},"alternative-id":["S0022481200049288"],"URL":"https:\/\/doi.org\/10.2307\/2273516","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,9]]}}}