{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T15:42:40Z","timestamp":1762270960379},"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":10054,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1986,9]]},"abstract":"<jats:p>Let <jats:italic>M<\/jats:italic> be an O-minimal structure. We use our understanding, acquired in [KPS], of the structure of definable sets of <jats:italic>n<\/jats:italic>-tuples in <jats:italic>M<\/jats:italic>, to study definable (in <jats:italic>M<\/jats:italic>) equivalence relations on <jats:italic>M<jats:sup>n<\/jats:sup><\/jats:italic>. In particular, we show that if <jats:italic>E<\/jats:italic> is an <jats:italic>A<\/jats:italic>-definable equivalence relation on <jats:italic>M<jats:sup>n<\/jats:sup><\/jats:italic> (<jats:italic>A<\/jats:italic> \u2282 <jats:italic>M<\/jats:italic>) then <jats:italic>E<\/jats:italic> has only finitely many classes with nonempty interior in <jats:italic>M<jats:sup>n<\/jats:sup><\/jats:italic>, each such class being moreover also <jats:italic>A<\/jats:italic>-definable. As a consequence, we are able to give some conditions under which an <jats:italic>O<\/jats:italic>-minimal theory <jats:italic>T eliminates imaginaries<\/jats:italic> (in the sense of Poizat [P]).<\/jats:p><jats:p>If <jats:italic>L<\/jats:italic> is a first order language and <jats:italic>M<\/jats:italic> an <jats:italic>L<\/jats:italic>-structure, then by a <jats:italic>definable set<\/jats:italic> in <jats:italic>M<\/jats:italic>, we mean something of the form <jats:italic>X<\/jats:italic> \u2282 <jats:italic>M<jats:sup>n<\/jats:sup><\/jats:italic>, <jats:italic>n<\/jats:italic> \u2265 1, where <jats:italic>X<\/jats:italic> = {(<jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>\u2026,<jats:italic>a<jats:sub>n<\/jats:sub><\/jats:italic>) \u2208 <jats:italic>M<jats:sup>n<\/jats:sup><\/jats:italic>: <jats:italic>M<\/jats:italic> \u22a8<jats:italic>\u03d5<\/jats:italic>(\u0101)} for some formula <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030875_inline1\" \/> \u2208 <jats:italic>L<\/jats:italic>(<jats:italic>M<\/jats:italic>). (Here <jats:italic>L<\/jats:italic>(<jats:italic>M<\/jats:italic>) means <jats:italic>L<\/jats:italic> together with names for the elements of <jats:italic>M<\/jats:italic>.) If the parameters from <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030875_inline1\" \/> come from a subset <jats:italic>A<\/jats:italic> of <jats:italic>M<\/jats:italic>, we say that <jats:italic>X<\/jats:italic> is <jats:italic>A-definable<\/jats:italic>.<\/jats:p><jats:p><jats:italic>M<\/jats:italic> is said to be <jats:italic>O-minimal<\/jats:italic> if <jats:italic>M<\/jats:italic> = (<jats:italic>M<\/jats:italic>, &lt;,\u2026), where &lt; is a dense linear order with no first or last element, and every definable set <jats:italic>X<\/jats:italic> \u2282 <jats:italic>M<\/jats:italic> is a finite union of points, and intervals (<jats:italic>a, b<\/jats:italic>) (where <jats:italic>a, b<\/jats:italic> \u2208 <jats:italic>M<\/jats:italic> \u222a {\u00b1 \u221e}). (This notion is as in [PS] except here we demand the underlying order be dense.) The complete theory <jats:italic>T<\/jats:italic> is said to be <jats:italic>O-minimal<\/jats:italic> if every model of <jats:italic>T<\/jats:italic> is O-minimal. (Note that in [KPS] it is proved that if <jats:italic>M<\/jats:italic> is O-minimal, then <jats:italic>T<\/jats:italic> = Th(<jats:italic>M<\/jats:italic>) is O-minimal.) In the remainder of this section and in \u00a72, M <jats:italic>will denote a fixed but arbitrary O-minimal structure. A,B,C,\u2026 will denote subsets of M<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2274024","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:18:32Z","timestamp":1146953912000},"page":"709-714","source":"Crossref","is-referenced-by-count":17,"title":["Some remarks on definable equivalence relations in O-minimal structures"],"prefix":"10.1017","volume":"51","author":[{"given":"Anand","family":"Pillay","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200030875_ref001","volume-title":"Transactions of the American Mathematical Society","author":"Knight"},{"key":"S0022481200030875_ref006","volume-title":"Classification theory and the number of nonisomorphic models","author":"Shelah","year":"1978"},{"key":"S0022481200030875_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71813-5"},{"key":"S0022481200030875_ref007","first-page":"625","volume":"49","author":"van den Dries","year":"1984","journal-title":"Algebraic theories with definable Skolem functions"},{"key":"S0022481200030875_ref002","first-page":"249","volume":"47","author":"Lascar","year":"1982","journal-title":"On the category of models of a complete theory"},{"key":"S0022481200030875_ref004","first-page":"1151","volume":"48","author":"Poizat","year":"1983","journal-title":"Une th\u00e9orie de Galois imaginaire"},{"key":"S0022481200030875_ref005","volume-title":"Transactions of the American Mathematical Society","author":"Pillay"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200030875","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T04:40:51Z","timestamp":1558500051000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200030875\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,9]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1986,9]]}},"alternative-id":["S0022481200030875"],"URL":"https:\/\/doi.org\/10.2307\/2274024","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,9]]}}}