{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T06:34:07Z","timestamp":1780382047604,"version":"3.54.1"},"reference-count":16,"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:title>Abstract<\/jats:title><jats:p>A Boolean algebra <jats:italic>B<\/jats:italic> is said to be <jats:italic>openly generated<\/jats:italic> if {<jats:italic>A<\/jats:italic> : <jats:italic>A<\/jats:italic> \u2264<jats:sub>rc<\/jats:sub><jats:italic>B<\/jats:italic>, \u2223<jats:italic>A<\/jats:italic>\u2223 = \u2135<jats:sub>0<\/jats:sub>} includes a club subset of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200020405_inline1\"\/>. We show:<\/jats:p><jats:p>(<jats:italic>V<\/jats:italic> = <jats:italic>L<\/jats:italic>). For any cardinal <jats:italic>\u03ba<\/jats:italic> there exists an <jats:italic>L<\/jats:italic><jats:sub>\u221e<jats:italic>\u03ba<\/jats:italic><\/jats:sub>-free Boolean algebra which is not openly generated (Proposition 4.1).<\/jats:p><jats:p>(MA+(<jats:italic>\u03c3<\/jats:italic>-closed)). Every <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200020405_inline2\"\/>-free Boolean algebra is openly generated (Theorem 4.2).<\/jats:p><jats:p>The last assertion follows from a characterization of openly generated Boolean algebras under MA<jats:sup>+<\/jats:sup>(<jats:italic>\u03c3<\/jats:italic>-closed) (Theorem 3.1). Using this characterization we also prove the independence of problem 7 in \u0160\u010depin [15] (Proposition 4.3 and Theorem 4.4).<\/jats:p>","DOI":"10.2307\/2275267","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:51:08Z","timestamp":1146941468000},"page":"302-310","source":"Crossref","is-referenced-by-count":4,"title":["Some remarks on openly generated Boolean algebras"],"prefix":"10.1017","volume":"59","author":[{"given":"Saka\u00e9","family":"Fuchino","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200020405_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0090944"},{"key":"S0022481200020405_ref013","first-page":"741","volume-title":"Handbook of Boolean Algebras","volume":"3","author":"Koppelberg","year":"1989"},{"key":"S0022481200020405_ref012","volume-title":"Handbook of Boolean Algebras","volume":"1","author":"Koppelberg","year":"1989"},{"key":"S0022481200020405_ref011","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511721168"},{"key":"S0022481200020405_ref010","unstructured":"[10] Heindorf L. , Openly generated Boolean algebras, mimeograph dated on 10 9, 1992."},{"key":"S0022481200020405_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(92)90038-2"},{"key":"S0022481200020405_ref008","volume-title":"Proceedings of the 10th Easter Conference","author":"Fuchino","year":"1993"},{"key":"S0022481200020405_ref006","article-title":"Martin's maximum, saturated ideals, and nonregular ultrafilters. part I","volume":"127","author":"Foreman","year":"1988","journal-title":"Annals of Mathematics"},{"key":"S0022481200020405_ref005","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-58.2.237"},{"key":"S0022481200020405_ref004","first-page":"423","volume-title":"Model-theoretic logics","author":"Eklov","year":"1985"},{"key":"S0022481200020405_ref016","first-page":"360","volume":"52","author":"Shelah","year":"1987","journal-title":"Semiproper forcing axiom implies Martin's maximum but not PFA+"},{"key":"S0022481200020405_ref003","volume-title":"Handbook of set theoretic topology","author":"Baumgartner","year":"1984"},{"key":"S0022481200020405_ref015","first-page":"3","article-title":"Functors and uncountable powers of compacta","volume":"36","author":"\u0160\u010depin","year":"1981","journal-title":"Uspechi Matematicheskikh Nauk"},{"key":"S0022481200020405_ref007","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093634482"},{"key":"S0022481200020405_ref002","volume-title":"Model-theoretic logics","author":"Barwise","year":"1985"},{"key":"S0022481200020405_ref001","first-page":"196","article-title":"Factorization theorems and \u2018strong\u2019 sequences in bicompacta","volume":"22","author":"Bandlow","year":"1980","journal-title":"Soviet Mathematics Doklady"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200020405","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T15:16:43Z","timestamp":1557933403000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200020405\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,3]]},"references-count":16,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1994,3]]}},"alternative-id":["S0022481200020405"],"URL":"https:\/\/doi.org\/10.2307\/2275267","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,3]]}}}