{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T06:18:31Z","timestamp":1649053111580},"reference-count":6,"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":8412,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1991,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>M<\/jats:italic> be a given model with similarity type <jats:italic>L<\/jats:italic> = <jats:italic>L<\/jats:italic>(<jats:italic>M<\/jats:italic>), and let <jats:italic>L<\/jats:italic>\u2032 be any fragment of <jats:italic>L<\/jats:italic><jats:sub>\u2223<jats:italic>L<\/jats:italic>(<jats:italic>M<\/jats:italic>\u2223<jats:sup>+<\/jats:sup>,<jats:italic>\u03c9<\/jats:italic><\/jats:sub> of cardinality \u2223<jats:italic>L<\/jats:italic>(<jats:italic>M<\/jats:italic>)\u2223. We call <jats:italic>N<\/jats:italic> \u227a <jats:italic>M<\/jats:italic><jats:italic>L<\/jats:italic>\u2032-<jats:italic>relatively saturated<\/jats:italic> iff for every <jats:italic>B<\/jats:italic> \u2286 <jats:italic>N<\/jats:italic> of cardinality less than \u2225<jats:italic>N<\/jats:italic>\u2225 every <jats:italic>L<\/jats:italic>\u2032-type over <jats:italic>B<\/jats:italic> which is realized in <jats:italic>M<\/jats:italic> is realized in <jats:italic>N<\/jats:italic>. We discuss the existence of such submodels.<\/jats:p><jats:p>The following are corollaries of the existence theorems.<\/jats:p><jats:p>(1) If <jats:italic>M<\/jats:italic> is of cardinality at least \u2136<jats:sub><jats:italic>\u03c9<\/jats:italic>1<\/jats:sub>, and fails to have the <jats:italic>\u03c9<\/jats:italic> order property, then there exists <jats:italic>N<\/jats:italic> \u227a <jats:italic>M<\/jats:italic> which is relatively saturated in <jats:italic>M<\/jats:italic> of cardinality \u2136<jats:sub><jats:italic>\u03c9<\/jats:italic>1<\/jats:sub>.<\/jats:p><jats:p>(2) Assume GCH. Let <jats:italic>\u03c8<\/jats:italic> \u2208 <jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic>1, <jats:italic>\u03c9<\/jats:italic><\/jats:sub>, and let <jats:italic>L<\/jats:italic>\u2032 \u2286 <jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic>1, <jats:italic>\u03c9<\/jats:italic><\/jats:sub> be a countable fragment containing <jats:italic>\u03c8<\/jats:italic>. If \u2203<jats:sub><jats:italic>\u03c7<\/jats:italic><\/jats:sub> &gt; \u2135<jats:sub>0<\/jats:sub> such that <jats:italic>I<\/jats:italic>(<jats:italic>\u03c7, \u03c8<\/jats:italic>) &lt; 2<jats:sup><jats:italic>\u03c7<\/jats:italic><\/jats:sup>, then for every <jats:italic>M<\/jats:italic> \u22a8 <jats:italic>\u03c8<\/jats:italic> and every cardinal <jats:italic>\u03bb<\/jats:italic> &lt; \u2225<jats:italic>M<\/jats:italic>\u2225 of uncountable cofinality, <jats:italic>M<\/jats:italic> has an <jats:italic>L<\/jats:italic>\u2032-relatively saturated submodel of cardinality <jats:italic>\u03bb<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2274909","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:38:25Z","timestamp":1146955105000},"page":"124-128","source":"Crossref","is-referenced-by-count":3,"title":["On chains of relatively saturated submodels of a model without the order property"],"prefix":"10.1017","volume":"56","author":[{"given":"Rami","family":"Grossberg","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200024956_ref006","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1972.41.247"},{"key":"S0022481200024956_ref002","first-page":"115","volume":"56","author":"Grossberg","year":"1991","journal-title":"Indiscernible sequences in a model which fails to have the order property"},{"key":"S0022481200024956_ref003","first-page":"302","volume":"51","author":"Grossberg","year":"1966","journal-title":"The number of nonisomorphic models of an infinitary theory which has the order property. Part A"},{"key":"S0022481200024956_ref001","first-page":"1292","volume":"55","author":"Albert","year":"1990","journal-title":"Rich models"},{"key":"S0022481200024956_ref005","first-page":"264","volume-title":"Classification theory (proceedings, Chicago, 1985","volume":"1292","author":"Shelah","year":"1988"},{"key":"S0022481200024956_ref004","volume-title":"Classification theory and the number of non-isomorphic models","author":"Shelah","year":"1978"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200024956","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T22:36:31Z","timestamp":1558132591000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200024956\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1991,3]]},"references-count":6,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1991,3]]}},"alternative-id":["S0022481200024956"],"URL":"https:\/\/doi.org\/10.2307\/2274909","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1991,3]]}}}