{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T04:45:50Z","timestamp":1772426750202,"version":"3.50.1"},"reference-count":13,"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":3388,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2004,12]]},"abstract":"<jats:title>Abstract.<\/jats:title><jats:p>We study the expansion of stable structures by adding predicates for <jats:italic>arbitrary<\/jats:italic> subsets. Generalizing work of Poizat-Bouscaren on the one hand and Baldwin-Benedikt-Casanovas-Ziegler on the other we provide a sufficient condition (Theorem 4.7) for such an expansion to be stable. This generalization weakens the original definitions in two ways: dealing with arbitrary subsets rather than just submodels and removing the \u2018small\u2019 or \u2018belles paires\u2019 hypothesis. We use this generalization to characterize in terms of pairs, the \u2018triviality\u2019 of the geometry on a strongly minimal set (Theorem 2.5). Call a set <jats:italic>A benign<\/jats:italic> if any type over <jats:italic>A<\/jats:italic> in the expanded language is determined by its restriction to the base language. We characterize the notion of benign as a kind of local homogenity (Theorem 1.7). Answering a question of [8] we characterize the property that <jats:italic>M<\/jats:italic> has the finite cover property over <jats:italic>A<\/jats:italic> (Theorem 3.9).<\/jats:p>","DOI":"10.2178\/jsl\/1102022221","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T21:52:14Z","timestamp":1109800334000},"page":"1243-1260","source":"Crossref","is-referenced-by-count":4,"title":["Local homogeneity"],"prefix":"10.1017","volume":"69","author":[{"given":"Bektur","family":"Baizhanov","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"John T.","family":"Baldwin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200007490_ref013","volume-title":"Classification Theory and the Number of Nonisomorphic Models","author":"Shelah","year":"1978"},{"key":"S0022481200007490_ref012","first-page":"239\u2013249","volume":"48","author":"Poizat","year":"1983","journal-title":"Paires de structure stables"},{"key":"S0022481200007490_ref008","first-page":"1127\u20131140","volume":"66","author":"Casanovas","year":"2001","journal-title":"Stable theories with a new predicate"},{"key":"S0022481200007490_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(89)90001-8"},{"key":"S0022481200007490_ref005","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093870870"},{"key":"S0022481200007490_ref004","first-page":"371\u2013391","volume":"65","author":"Baldwin","year":"2000","journal-title":"Constructing \u03c9-stable structures: Rank 2 fields"},{"key":"S0022481200007490_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-00-02672-6"},{"key":"S0022481200007490_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-07330-8"},{"key":"S0022481200007490_ref001","unstructured":"Baizhanov B. , Baldwin J.T. , and Shelah S. , Subsets of superstable structures are weakly benign, submitted."},{"key":"S0022481200007490_ref010","volume-title":"Stability in Model Theory","author":"Lascar","year":"1987"},{"key":"S0022481200007490_ref007","first-page":"1184\u20131194","volume":"56","author":"Buechler","year":"1991","journal-title":"Pseudoprojective strongly minimal sets are locally projective"},{"key":"S0022481200007490_ref009","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511551574"},{"key":"S0022481200007490_ref011","volume-title":"An introduction to stability theory","author":"Pillay","year":"1983"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200007490","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T19:39:26Z","timestamp":1557171566000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200007490\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,12]]},"references-count":13,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2004,12]]}},"alternative-id":["S0022481200007490"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1102022221","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,12]]}}}