{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T09:42:15Z","timestamp":1775209335453,"version":"3.50.1"},"reference-count":6,"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>The notion of forking has been introduced by Shelah, and a full treatment of it will appear in his book on stability [S1]. The principal aim of this paper is to show that it is an easy and natural notion.<\/jats:p><jats:p>Consider some well-known examples of \u2135<jats:sub>0<\/jats:sub>-stable theories: vector spaces over <jats:italic>Q<\/jats:italic>, algebraically closed fields, differentially closed fields of characteristic 0; in each of these cases, we have a natural notion of independence: linear, algebraic and differential independence respectively. Forking gives a generalization of these notions. More precisely, if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004826X_inline1\"\/> are subsets of some model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004826X_inline2\"\/> and <jats:italic>c<\/jats:italic> a point of this model, the fact that the type of <jats:italic>c<\/jats:italic> over <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004826X_inline3\"\/> does not fork over <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004826X_inline4\"\/> means that there are no more relations of dependence between <jats:italic>c<\/jats:italic> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004826X_inline3\"\/> than there already existed between <jats:italic>c<\/jats:italic> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004826X_inline4\"\/>. In the case of the vector spaces, this means that <jats:italic>c<\/jats:italic> is in the space generated by <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004826X_inline3\"\/> only if it is already in the space generated by <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004826X_inline4\"\/>. In the case of differentially closed fields, this means that the minimal differential equations of <jats:italic>c<\/jats:italic> with coefficient respectively in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004826X_inline4\"\/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004826X_inline3\"\/> have the same order. Of course, these notions of dependence are essential for the study of the above mentioned structures. Forking is no less important for stable theories. A glance at Shelah's book will convince the reader that this is the case.<\/jats:p><jats:p>What we have to do is the following. Assuming <jats:italic>T<\/jats:italic> stable and given <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004826X_inline1\"\/> and <jats:italic>p<\/jats:italic> a type on <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004826X_inline4\"\/>, we want to distinguish among the extensions of <jats:italic>p<\/jats:italic> to <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004826X_inline3\"\/> some of\nthem that we shall call the nonforking extensions of <jats:italic>p<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2273127","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:50:56Z","timestamp":1146952256000},"page":"330-350","source":"Crossref","is-referenced-by-count":61,"title":["An introduction to forking"],"prefix":"10.1017","volume":"44","author":[{"given":"Daniel","family":"Lascar","sequence":"first","affiliation":[]},{"given":"Bruno","family":"Poizat","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120004826X_ref005","volume-title":"Studies in logic and the foundations of mathematics","volume":"92","author":"Shelah","year":"1978"},{"key":"S002248120004826X_ref001","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S002248120004826X_ref003","first-page":"489","volume-title":"Logique et analyse","author":"Lascar","year":"1971"},{"key":"S002248120004826X_ref004","unstructured":"Poizat Bruno , D\u00e9viation des types, Doctoral dissertation, Paris, 1977."},{"key":"S002248120004826X_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF02771574"},{"key":"S002248120004826X_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757234"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120004826X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T20:53:08Z","timestamp":1558903988000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120004826X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,9]]},"references-count":6,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1979,9]]}},"alternative-id":["S002248120004826X"],"URL":"https:\/\/doi.org\/10.2307\/2273127","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,9]]}}}