{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,29]],"date-time":"2026-07-29T00:19:54Z","timestamp":1785284394450,"version":"3.55.0"},"reference-count":22,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":15259,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1972,6]]},"abstract":"<jats:p>One of the major problems of model theory is the spectrum problem, i.e. the development of structure theorems for the spectrum of models of a given theory. I hope that this paper will make a (small) contribution to the solution of this problem.<\/jats:p><jats:p>Broadly speaking, in this paper we take a fixed (but arbitrary) theory T and consider a particular class \u2130 of models of T. The structures in \u2130 (which are known as existentially closed structures) are connected with the model complete extensions of T. These structures have already appeared several times in the literature.<\/jats:p><jats:p>In \u00a71 we survey the notation and terminology that we use, as well as the well known facts that we require. We also consider several concepts which are not new here but which may not be well known.<\/jats:p><jats:p>In \u00a72 we define, and give the basic results concerning existentially closed structures. For most theories, T, the class \u2130 is not elementary and hence is not directly amenable to a model theoretic study. Because of this we consider a smaller class of structures\u2014the uniformly existentially closed structures. These are discussed in \u00a73.<\/jats:p><jats:p>In \u00a74 we look at model complete theories via existentially closed structures. This section is essentially a refinement of parts of [6].<\/jats:p><jats:p>In \u00a75 we look at model companions of theories via existentially closed structures.<\/jats:p><jats:p>Finally, in \u00a76 we make some remarks on the relevance of existentially closed structures to the spectrum problem.<\/jats:p>","DOI":"10.2307\/2272974","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:16:08Z","timestamp":1146950168000},"page":"293-310","source":"Crossref","is-referenced-by-count":24,"title":["Existentially closed structures"],"prefix":"10.1017","volume":"37","author":[{"given":"H.","family":"Simmons","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200079925_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90008-2"},{"key":"S0022481200079925_ref005","unstructured":"Fisher E. , letter to A. Macintyre."},{"key":"S0022481200079925_ref019","unstructured":"Simmons H. , The use of injective-like structures in model theory (submitted)."},{"key":"S0022481200079925_ref012","first-page":"287","volume-title":"Essays on the foundations of mathematics","author":"Rabin","year":"1962"},{"key":"S0022481200079925_ref018","volume-title":"Mathematical logic","author":"Shoenfield","year":"1967"},{"key":"S0022481200079925_ref007","volume-title":"Annals of Mathematical Logic","author":"Macintyre"},{"key":"S0022481200079925_ref014","first-page":"69","article-title":"Forcing in model theory","volume":"5","author":"Robinson","year":"1970","journal-title":"Symposia Mathematica"},{"key":"S0022481200079925_ref002","volume-title":"Models and ultraproducts","author":"Bell","year":"1969"},{"key":"S0022481200079925_ref003","first-page":"61","volume":"32","author":"Chang","year":"1967","journal-title":"Omitting types of prenex formulas"},{"key":"S0022481200079925_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90016-7"},{"key":"S0022481200079925_ref021","first-page":"137","article-title":"Structures pr\u00e9homog\u00e8nes et structures g\u00e9n\u00e9rigues","volume":"273","author":"Cusin","year":"1971","journal-title":"Comptes Rendus"},{"key":"S0022481200079925_ref006","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1964.tb01088.x"},{"key":"S0022481200079925_ref009","volume-title":"Proceedings of the conference on decision problems in group theory","author":"Neumann","year":"1971"},{"key":"S0022481200079925_ref010","first-page":"185","volume-title":"Studies in pure mathematics","author":"Neumann","year":"1971"},{"key":"S0022481200079925_ref022","first-page":"956","article-title":"Recherche du forcing-compagnon et du mod\u00e8le-compagnon d'une theoric li\u00e9e a l'existence de mod\u00e8les \u2135\u03b1-universels","volume":"273","author":"Cusin","year":"1971","journal-title":"Comptes Rendus"},{"key":"S0022481200079925_ref011","first-page":"151","volume-title":"Proceedings of the International Congress on logic, methodology and philosophy of science","author":"Rabin","year":"1962"},{"key":"S0022481200079925_ref013","volume-title":"Introduction to model theory and to the metamathematics of algebra","author":"Robinson","year":"1963"},{"key":"S0022481200079925_ref008","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-27.2.247"},{"key":"S0022481200079925_ref015","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)70851-6"},{"key":"S0022481200079925_ref016","first-page":"441","volume":"36","author":"Robinson","year":"1971","journal-title":"On the notion of algebraic closedness for noncommutative groups and fields"},{"key":"S0022481200079925_ref017","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1951-0040299-6"},{"key":"S0022481200079925_ref020","first-page":"1620","article-title":"Structures g\u00e9n\u00e9riques associ\u00e9es \u00e0 une classe de theories","volume":"272","author":"Cusin","year":"1971","journal-title":"Comptes Rendus"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200079925","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T21:14:26Z","timestamp":1559250866000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200079925\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1972,6]]},"references-count":22,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1972,6]]}},"alternative-id":["S0022481200079925"],"URL":"https:\/\/doi.org\/10.2307\/2272974","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1972,6]]}}}