{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,28]],"date-time":"2025-11-28T12:07:16Z","timestamp":1764331636512},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":10693,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1984,12]]},"abstract":"<jats:p>An impressive theory has been developed, largely by Shelah, around the notion of a stable theory. This includes detailed structure theorems for the models of such theories as well as a generalized notion of independence. The various stability properties can be defined in terms of the numbers of types over sets, or in terms of the complexity of definable sets. In the concrete examples of stable theories, however, one finds an important distinction between \u201cpositive\u201d and \u201cnegative\u201d information, such a distinction not being an a priori consequence of the general definitions. In the naive examples this may take the form of distinguishing between say a class of a definable equivalence relation and the complement of a class. In the more algebraic examples, this distinction may have a \u201ctopological\u201d significance, for example with the Zariski topology on (the set of <jats:italic>n<\/jats:italic>-tuples of) an algebraically closed field, the \u201cclosed\u201d sets being those given by sets of polynomial equalities. Note that in the latter case, every definable set is a Boolean combination of such closed sets (the definable sets are precisely the constructible sets). Similarly, stability conditions in practice reduce to chain conditions on certain \u201cspecial\u201d definable sets (e.g. in modules, stable groups). The aim here is to develop and present such notions in the general (model-theoretic) context. The basic notion is that of an \u201cequation\u201d. Given a complete theory <jats:italic>T<\/jats:italic> in a language <jats:italic>L<\/jats:italic>, an <jats:italic>L<\/jats:italic>-formula <jats:italic>\u03c6<\/jats:italic>(<jats:italic>x\u0304, \u0233<\/jats:italic>) is said to be an <jats:italic>equation<\/jats:italic> (in <jats:italic>x\u0304<\/jats:italic>) if any collection <jats:italic>\u03a6<\/jats:italic> of instances of <jats:italic>\u03c6<\/jats:italic>(i.e. of formulae <jats:italic>\u03c6<\/jats:italic>(<jats:italic>x\u0304, \u0101<\/jats:italic>)) is equivalent to a finite subset <jats:italic>\u03a6\u2032<\/jats:italic> \u2282 <jats:italic>\u03a6<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2274284","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:10:48Z","timestamp":1146953448000},"page":"1350-1362","source":"Crossref","is-referenced-by-count":13,"title":["Closed sets and chain conditions in stable theories"],"prefix":"10.1017","volume":"49","author":[{"given":"Anand","family":"Pillay","sequence":"first","affiliation":[]},{"given":"Gabriel","family":"Srour","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200042596_ref015","unstructured":"Srour G. , Equations and equational theories (in preparation)."},{"key":"S0022481200042596_ref014","unstructured":"Srour G. , Some clarifications to the theory of modules, preprints, Hebrew University, Jerusalem, 1981\/81. (Extensions of .)"},{"key":"S0022481200042596_ref011","volume-title":"Introduction to stability theory","author":"Pillay","year":"1983"},{"key":"S0022481200042596_ref010","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1983-0718994-3"},{"key":"S0022481200042596_ref009","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-46.2.365"},{"key":"S0022481200042596_ref008","first-page":"330","volume":"44","author":"and","year":"1979","journal-title":"Introduction to forking"},{"key":"S0022481200042596_ref007","first-page":"153","volume-title":"Set theory and hierarchy theory","volume":"537","author":"Lachlan","year":"1976"},{"key":"S0022481200042596_ref003","first-page":"53","volume-title":"Logic Colloquium '78","author":"Cherlin","year":"1979"},{"key":"S0022481200042596_ref002","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200042596_ref013","unstructured":"Srour G. , Some immediate clarifications to the theory of modules, preprint, Hebrew University, Jerusalem, 1981."},{"key":"S0022481200042596_ref016","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(84)90014-9"},{"key":"S0022481200042596_ref012","volume-title":"Classification theory and the number of nonisomorphic models","author":"Shelah","year":"1978"},{"key":"S0022481200042596_ref004","unstructured":"Cherlin G. , Harrington L. and Lachlan A. H. , \u21350-categorical, \u21350-stable structures (to appear)."},{"key":"S0022481200042596_ref005","unstructured":"Harrington L. and Harnik V. , Fundamentals of forking (to appear)."},{"key":"S0022481200042596_ref001","volume-title":"Groupe d'\u00c9tude: Th\u00e9ories Stables, 3e annee: 1979\/80","author":"Berline","year":"1982"},{"key":"S0022481200042596_ref006","doi-asserted-by":"publisher","DOI":"10.4064\/fm-81-2-133-145"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200042596","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T19:10:13Z","timestamp":1558638613000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200042596\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1984,12]]},"references-count":16,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1984,12]]}},"alternative-id":["S0022481200042596"],"URL":"https:\/\/doi.org\/10.2307\/2274284","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1984,12]]}}}