{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,7]],"date-time":"2025-07-07T10:04:34Z","timestamp":1751882674755},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12337,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1980,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Theorem 1. (\u25ca<jats:sub>\u21351<\/jats:sub>,) <jats:italic>If B is an infinite Boolean algebra (BA), then there is B<jats:sub>1<\/jats:sub>, such that<\/jats:italic> \u2223 Aut (<jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub>) \u2264\u2223<jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub>\u2223 = \u2135<jats:sub>1<\/jats:sub> and \u3008<jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub>, Aut (B<jats:sub>1<\/jats:sub>)\u3009 \u2261 \u3008<jats:italic>B<\/jats:italic>, Aut(B)\u3009.<\/jats:p><jats:p>Theorem 2. (\u25ca<jats:sub>\u21351<\/jats:sub>) <jats:italic>There is a countably compact logic stronger than first-order logic even on finite models<\/jats:italic>.<\/jats:p><jats:p>This partially answers a question of H. Friedman. These theorems appear in \u00a7\u00a71 and 2.<\/jats:p><jats:p>Theorem 3. (a) (\u25ca<jats:sub>\u21351<\/jats:sub>) <jats:italic>If B is an atomic \u2135-saturated infinite BA<\/jats:italic>, \u03a8 \u0404 L<jats:sub>\u03c91\u03c9<\/jats:sub> and \u3008<jats:italic>B<\/jats:italic>, Aut (<jats:italic>B<\/jats:italic>)\u3009 \u22a8\u03a8 <jats:italic>then there is B<jats:sub>1<\/jats:sub>, Such that<\/jats:italic> \u2223Aut(<jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub>)\u2223 \u2264 \u2223<jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub>\u2223 =\u2135<jats:sub>1<\/jats:sub>, <jats:italic>and<\/jats:italic> \u3008<jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub>, Aut(<jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub>)\u3009\u22a8\u03a8. <jats:italic>In particular if B is 1-homogeneous so is B<\/jats:italic><jats:sub>1<\/jats:sub>. (b) (a) <jats:italic>holds for B = P(\u03c9) even if we assume only CH<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2273187","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:52:48Z","timestamp":1146937968000},"page":"265-283","source":"Crossref","is-referenced-by-count":9,"title":["On the elementary equivalence of automorphism groups of Boolean algebras; downward Skolem L\u00f6wenheim theorems and compactness of related quantifiers"],"prefix":"10.1017","volume":"45","author":[{"given":"Matatyahu","family":"Rubin","sequence":"first","affiliation":[]},{"given":"Saharon","family":"Shelah","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200046788_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF02762670"},{"key":"S0022481200046788_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BF02488016"},{"key":"S0022481200046788_ref002","first-page":"331","volume-title":"Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics","volume":"XXV","author":"Keisler","year":"1974"},{"key":"S0022481200046788_ref005","article-title":"On the automorphism groups of countable Boolean algebras","author":"Rubin","journal-title":"Israel Journal of Mathematics"},{"key":"S0022481200046788_ref001","first-page":"113","volume":"40","author":"Friedman","year":"1975","journal-title":"One hundred and two problems in mathematical logic"},{"key":"S0022481200046788_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90008-6"},{"key":"S0022481200046788_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF02944950"},{"key":"S0022481200046788_ref008","unstructured":"Shelah S. , Models with second order properties IV (preprint)."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200046788","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T15:12:19Z","timestamp":1558883539000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200046788\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1980,6]]},"references-count":8,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1980,6]]}},"alternative-id":["S0022481200046788"],"URL":"https:\/\/doi.org\/10.2307\/2273187","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1980,6]]}}}