{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T17:15:05Z","timestamp":1784308505299,"version":"3.55.0"},"reference-count":7,"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":4575,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2001,9]]},"abstract":"<jats:p>Let <jats:italic>M<\/jats:italic> be an <jats:italic>L<\/jats:italic>-structure and <jats:italic>A<\/jats:italic> be an infinite subset of <jats:italic>M<\/jats:italic>. Two structures can be defined from <jats:italic>A<\/jats:italic>:<\/jats:p><jats:p>\u2022 The <jats:italic>induced<\/jats:italic> structure on <jats:italic>A<\/jats:italic> has a name R<jats:sub><jats:italic>\u03c6<\/jats:italic><\/jats:sub> for every \u2205-definable relation <jats:italic>\u03c6<\/jats:italic>(<jats:italic>M<\/jats:italic>) \u2229 <jats:italic>A<jats:sup>n<\/jats:sup><\/jats:italic> on <jats:italic>A<\/jats:italic>. Its language is<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200010537_Uequ1\"\/><\/jats:disp-formula><\/jats:p><jats:p><jats:italic>A<\/jats:italic> with its L<jats:sub>ind<\/jats:sub>-structure will be denoted by <jats:italic>A<\/jats:italic><jats:sub>ind<\/jats:sub>.<\/jats:p><jats:p>\u2022 The pair (<jats:italic>M, A<\/jats:italic>) is an <jats:italic>L(P)<\/jats:italic>-structure, where <jats:italic>P<\/jats:italic> is a unary predicate for <jats:italic>A<\/jats:italic> and <jats:italic>L(P)<\/jats:italic> = <jats:italic>L<\/jats:italic> \u222a{<jats:italic>P<\/jats:italic>}.<\/jats:p><jats:p>We call <jats:italic>A small<\/jats:italic> if there is a pair (<jats:italic>N, B<\/jats:italic>) elementarily equivalent to (<jats:italic>M, A<\/jats:italic>) and such that for every finite subset <jats:italic>b<\/jats:italic> of <jats:italic>N<\/jats:italic> every <jats:italic>L<\/jats:italic>\u2013type over <jats:italic>Bb<\/jats:italic> is realized in <jats:italic>N<\/jats:italic>.<\/jats:p><jats:p>A formula <jats:italic>\u03c6<\/jats:italic>(<jats:italic>x, y<\/jats:italic>) has the <jats:italic>finite cover property<\/jats:italic> (f.c.p) in <jats:italic>M<\/jats:italic> if for all natural numbers <jats:italic>k<\/jats:italic> there is a set of <jats:italic>\u03c6<\/jats:italic>\u2013formulas<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200010537_Uequ2\"\/><\/jats:disp-formula><\/jats:p><jats:p>which is <jats:italic>k<\/jats:italic>\u2013consistent but not consistent in <jats:italic>M. M<\/jats:italic> has the f.c.p if some formula has the f.c.p in <jats:italic>M<\/jats:italic>. It is well known that unstable structures have the f.c.p. (see [6].) We will prove the following two theorems.<\/jats:p><jats:p>Theorem A. <jats:italic>Let A be a small subset of M. If M does not have the finite cover property then, for every<\/jats:italic> \u03bb \u2265 \u2223<jats:italic>L<\/jats:italic>\u2223, <jats:italic>if both M and<\/jats:italic><jats:italic>A<\/jats:italic><jats:sub>ind<\/jats:sub><jats:italic>are<\/jats:italic> \u03bb\u2013<jats:italic>stable then (M, A) is<\/jats:italic> \u03bb\u2013<jats:italic>stable<\/jats:italic>.<\/jats:p><jats:p>Corollary 1.1 (Poizat [5]). <jats:italic>If M does not have the finite cover property and N<\/jats:italic> \u227a <jats:italic>M is a small elementary substructure, then (M, N) is stable<\/jats:italic>.<\/jats:p><jats:p>Corollary 1.2 (Zilber [7]). <jats:italic>If U is the group of wots of unity in the field<\/jats:italic> \u2102 <jats:italic>of complex numbers the pair<\/jats:italic> (\u2102, <jats:italic>U<\/jats:italic>) <jats:italic>is<\/jats:italic><jats:italic>\u03c9<\/jats:italic>\u2013<jats:italic>stable<\/jats:italic>.<\/jats:p><jats:p>Proof. (See [4].) As a strongly minimal set \u2102 is \u03c9\u2013stable and does not have the f.c.p. By the subspace theorem of Schmidt [3] every algebraic set intersects <jats:italic>U<\/jats:italic> in a finite union of translates of subgroups definable in the group structure of <jats:italic>U<\/jats:italic> alone. Whence <jats:italic>U<\/jats:italic><jats:sub>ind<\/jats:sub> is nothing more than a (divisible) abelian group, which is <jats:italic>\u03c9<\/jats:italic>\u2013stable.<\/jats:p>","DOI":"10.2307\/2695097","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:06:19Z","timestamp":1146938779000},"page":"1127-1140","source":"Crossref","is-referenced-by-count":23,"title":["Stable theories with a new predicate"],"prefix":"10.1017","volume":"66","author":[{"given":"Enrique","family":"Casanovas","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martin","family":"Ziegler","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200010537_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-00-02672-6"},{"key":"S0022481200010537_ref005","first-page":"239","volume":"48","author":"Poizat","year":"1983","journal-title":"Paires de structures stables"},{"key":"S0022481200010537_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-68521-0_6"},{"key":"S0022481200010537_ref006","volume-title":"Classification Theory","author":"Shelah","year":"1990"},{"key":"S0022481200010537_ref007","unstructured":"Zilber B. , Unpublished."},{"key":"S0022481200010537_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-48208-6_4"},{"key":"S0022481200010537_ref002","first-page":"434","volume":"53","author":"Bouscarbn","year":"1988","journal-title":"Des belles paires aux beaux uples"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200010537","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,7]],"date-time":"2019-05-07T23:26:27Z","timestamp":1557271587000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200010537\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,9]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2001,9]]}},"alternative-id":["S0022481200010537"],"URL":"https:\/\/doi.org\/10.2307\/2695097","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2001,9]]}}}