{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T12:11:17Z","timestamp":1648642277653},"reference-count":12,"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":15167,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1972,9]]},"abstract":"If \u03a3 is the class of all fields and \u03a3* is the class of all algebraically closed fields, then it is well known that \u03a3* is characterized by the following properties:<\/jats:p>(i) \u03a3* is the class of models of some first order theory K<\/jats:italic>*.<\/jats:p>(ii) If m<\/jats:italic>1<\/jats:sub>m<\/jats:italic>2<\/jats:sub> are in \u03a3* and m<\/jats:italic>1<\/jats:sub> \u2286 m<\/jats:italic>2<\/jats:sub> then m<\/jats:italic>1<\/jats:sub> \u227a m<\/jats:italic>2<\/jats:sub> (m<\/jats:italic>1<\/jats:sub> is an elementary substructure of m<\/jats:italic>2<\/jats:sub>, i.e. any first order sentence true in m<\/jats:italic>1<\/jats:sub> is true in m<\/jats:italic>2<\/jats:sub>).<\/jats:p>(iii) If m<\/jats:italic>1<\/jats:sub> is in \u03a3 then there is a structure m<\/jats:italic>2<\/jats:sub> in \u03a3* such that m<\/jats:italic>1<\/jats:sub> \u2286 m<\/jats:italic>2<\/jats:sub>.<\/jats:p>If \u03a3 is some other class of models of a first order theory K<\/jats:italic> and a subclass \u03a3* of \u03a3 exists satisfying (i)\u2013(iii) then \u03a3* is uniquely determined and K<\/jats:italic>* (which is unique up to logical equivalence) is called the model-companion<\/jats:italic> of K<\/jats:italic>. This notion is a generalization of the fundamental notion of model-completion introduced and extensively studied by A. Robinson [6], When the model-companion exists it provides the basis for a satisfactory treatment of the notion of an algebraically closed model of K<\/jats:italic>.<\/jats:p>Recently A. Robinson has developed a more general formulation of the notion of \u201calgebraically closed\u201d structures in \u03a3, which is applicable to any inductive elementary class \u03a3 of structures (by elementary we always mean EC\u0394<\/jats:sub>). Condition (i) must be weakened to<\/jats:p>(i\u2032) \u03a3* is closed under elementary substructure (i.e. if m<\/jats:italic>1<\/jats:sub> is in \u03a3* and m<\/jats:italic>2<\/jats:sub> \u227a m<\/jats:italic>1<\/jats:sub> then m<\/jats:italic>2<\/jats:sub> is in \u03a3*).<\/jats:p>","DOI":"10.2307\/2272742","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:17:55Z","timestamp":1146950275000},"page":"546-556","source":"Crossref","is-referenced-by-count":4,"title":["The model-companion of a class of structures"],"prefix":"10.1017","volume":"37","author":[{"given":"G. L.","family":"Cherlin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200079111_ref011","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1951-0040299-6"},{"key":"S0022481200079111_ref010","first-page":"668","article-title":"Infinite forcing in model theory and m-existentially closed structures","volume":"18","author":"Saracino","year":"1971","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200079111_ref009","first-page":"441","volume":"36","author":"Robinson","year":"1971","journal-title":"On the notion of algebraic closedness for noncommutative groups and fields"},{"key":"S0022481200079111_ref008","volume-title":"Forcing in model theory","author":"Robinson","year":"1970"},{"key":"S0022481200079111_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)70851-6"},{"key":"S0022481200079111_ref006","volume-title":"Introduction to model theory and to the metamathematics of algebra","author":"Robinson","year":"1963"},{"key":"S0022481200079111_ref001","unstructured":"Coven C. , Forcing in infinitary languages, Doctoral Dissertation, Yale University, New Haven, Ct., 1971."},{"key":"S0022481200079111_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90016-7"},{"key":"S0022481200079111_ref012","unstructured":"Cherlin G. , Second order forcing, algebraic completeness, and large cardinals (in preparation)."},{"key":"S0022481200079111_ref003","unstructured":"Fisher E. , Homogeneous universal models revisited (in preparation)."},{"key":"S0022481200079111_ref004","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19690150107"},{"key":"S0022481200079111_ref005","volume-title":"Languages with expressions of infinite length","author":"Karp","year":"1964"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200079111","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T20:58:13Z","timestamp":1559249893000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200079111\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1972,9]]},"references-count":12,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1972,9]]}},"alternative-id":["S0022481200079111"],"URL":"https:\/\/doi.org\/10.2307\/2272742","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1972,9]]}}}