{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T15:49:07Z","timestamp":1649000947849},"reference-count":10,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":7316,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1994,3]]},"abstract":"<jats:p>In this paper we study nonmultidimensional superstable theories <jats:italic>T<\/jats:italic>, possibly in an uncountable language, and develop some techniques permitting the generalisation of certain results from the finite rank (and\/or countable language) context to the general case.<\/jats:p><jats:p>We prove, among other things, the following: there is a set <jats:italic>A<\/jats:italic><jats:sub>0<\/jats:sub> of parameters, which has cardinality at most \u2223<jats:italic>T<\/jats:italic>\u2223, and in the finite-dimensional case is finite, such that over any <jats:italic>B<\/jats:italic> \u2287 <jats:italic>A<\/jats:italic><jats:sub>0<\/jats:sub> there is a locally atomic model. One of the consequences of this is that if <jats:bold>C<\/jats:bold> is the monster model of <jats:italic>T<\/jats:italic>, <jats:italic>\u03c6<\/jats:italic>(<jats:italic>x<\/jats:italic>) is a formula over <jats:italic>A<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>\u03c6<\/jats:italic><jats:sup><jats:bold>C<\/jats:bold><\/jats:sup> \u2287 <jats:italic>X<\/jats:italic> and (<jats:italic>X, \u03c6<\/jats:italic><jats:sup><jats:bold>C<\/jats:bold><\/jats:sup>) satisfies the Tarski-Vaught condition after adding names for <jats:italic>A<\/jats:italic><jats:sub>0<\/jats:sub>, then there is an elementary substructure <jats:italic>M<\/jats:italic> of <jats:bold>C<\/jats:bold> containing <jats:italic>A<\/jats:italic><jats:sub>0<\/jats:sub> such that <jats:italic>\u03c6<\/jats:italic><jats:sup><jats:italic>M<\/jats:italic><\/jats:sup> = <jats:italic>X<\/jats:italic>. Applications to the spectrum problem will appear in [Ch-P].<\/jats:p><jats:p>In fact, all the components of the machinery we develop are already present in the general theory. One such component involves a stratification of the regular types of <jats:italic>T<\/jats:italic> using a generalized notion of weakly minimal formula. This appears in [Sh, Chapter V and the proof of IX.2.4] and also in [P2]. A second component involves definable groups which arise as \u2018binding\u201d groups. The existence of such groups, under certain hypotheses on the behavior of nonorthogonality, is due to Hrushovski [Hr1], and our use of them to help obtain \u201cj-constructible\u201d models is similar to their use in [Bu-Sh].<\/jats:p>","DOI":"10.2307\/2275257","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:51:08Z","timestamp":1146955868000},"page":"151-165","source":"Crossref","is-referenced-by-count":1,"title":["Some remarks on nonmultidimensional superstable theories"],"prefix":"10.1017","volume":"59","author":[{"given":"Anand","family":"Pillay","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200020302_ref005","first-page":"207","volume":"54","author":"Hrushovski","year":"1989","journal-title":"Kueker's conjecture for stable theories"},{"key":"S0022481200020302_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(89)90059-6"},{"key":"S0022481200020302_ref002","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/0168-0072(89)90039-0","article-title":"On the existence of regular types","volume":"45","author":"Buechler","year":"1989","journal-title":"Annals of Pure and Applied Logic"},{"key":"S0022481200020302_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760649"},{"key":"S0022481200020302_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(90)90046-5"},{"key":"S0022481200020302_ref001","volume-title":"Fundamentals of stability theory","author":"Baldwin","year":"1987"},{"key":"S0022481200020302_ref003","unstructured":"Chowdhury A. and Pillay A. , On the number of models of uncountable theories, preprint (1993)."},{"key":"S0022481200020302_ref008","volume-title":"Introduction to stability theory","author":"Pillay","year":"1983"},{"key":"S0022481200020302_ref009","first-page":"880","volume":"49","author":"Pillay","year":"1984","journal-title":"Regular types in nonmultidimensional \u03c9-stable theories"},{"key":"S0022481200020302_ref010","volume-title":"Classification theory","author":"Shelah","year":"1990"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200020302","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T19:16:21Z","timestamp":1557947781000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200020302\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,3]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1994,3]]}},"alternative-id":["S0022481200020302"],"URL":"https:\/\/doi.org\/10.2307\/2275257","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,3]]}}}