{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T10:22:59Z","timestamp":1648549379663},"reference-count":15,"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":2202,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2008,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This article is devoted to two different generalizations of projective Boolean algebras: openly generated Boolean algebras and tightly <jats:italic>\u03c3<\/jats:italic>-filtered Boolean algebras.<\/jats:p><jats:p>We show that for every uncountable regular cardinal <jats:italic>\u03ba<\/jats:italic> there are 2<jats:sup><jats:italic>\u03ba<\/jats:italic><\/jats:sup> pairwise non-isomorphic openly generated Boolean algebras of size <jats:italic>\u03ba<\/jats:italic> &gt; \u2135<jats:sub>1<\/jats:sub> provided there is an almost free non-free abelian group of size <jats:italic>\u03ba<\/jats:italic>. The openly generated Boolean algebras constructed here are almost free.<\/jats:p><jats:p>Moreover, for every infinite regular cardinal <jats:italic>\u03ba<\/jats:italic> we construct 2<jats:sup><jats:italic>\u03ba<\/jats:italic><\/jats:sup> pairwise non-isomorphic Boolean algebras of size <jats:italic>\u03ba<\/jats:italic> that are tightly <jats:italic>\u03c3<\/jats:italic>-filtered and c.c.c.<\/jats:p><jats:p>These two results contrast nicely with Koppelberg's theorem in [12] that for every uncountable regular cardinal <jats:italic>\u03ba<\/jats:italic> there are only 2<jats:sup><jats:italic>\u03ba<\/jats:italic><\/jats:sup> isomorphism types of projective Boolean algebras of size <jats:italic>\u03ba<\/jats:italic>.<\/jats:p>","DOI":"10.2178\/jsl\/1208358746","type":"journal-article","created":{"date-parts":[[2008,10,14]],"date-time":"2008-10-14T15:19:02Z","timestamp":1223997542000},"page":"151-164","source":"Crossref","is-referenced-by-count":0,"title":["The number of openly generated Boolean algebras"],"prefix":"10.1017","volume":"73","author":[{"given":"Stefan","family":"Geschke","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Saharon","family":"Shelah","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200004655_ref008","volume-title":"Infinite abelian groups","volume":"2","author":"Fuchs","year":"1970"},{"key":"S0022481200004655_ref002","doi-asserted-by":"publisher","DOI":"10.1142\/S021906130100003X"},{"key":"S0022481200004655_ref003","volume-title":"Almost free modules: Set-theoretic methods","author":"Eklof","year":"1990"},{"key":"S0022481200004655_ref014","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1994-1249391-8"},{"key":"S0022481200004655_ref007","unstructured":"Fuchino S. and Shelah S. , More on l \u03ba\u221e-free Boolean algebras, in preparation."},{"key":"S0022481200004655_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/s00012-002-8176-1"},{"key":"S0022481200004655_ref012","first-page":"741","volume-title":"Handbook of boolean algebras","volume":"3","author":"Koppelberg","year":"1989"},{"key":"S0022481200004655_ref013","first-page":"119","volume-title":"Advances in algebra and model theory","author":"Koppelberg","year":"1997"},{"key":"S0022481200004655_ref001","doi-asserted-by":"publisher","DOI":"10.4064\/fm-78-3-193-203"},{"key":"S0022481200004655_ref015","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(02)00037-4"},{"key":"S0022481200004655_ref009","unstructured":"Geschke S. , On \u03c3-filtered Boolean algebras, Ph.D. thesis, Freie Universt\u00e4t Berlin, 2000."},{"key":"S0022481200004655_ref004","volume-title":"Almost free modules: Set-theoretic methods","volume":"65","author":"Eklof","year":"2002"},{"key":"S0022481200004655_ref005","volume-title":"Set-theoretic aspects of almost free Boolean algebras","author":"Fuchino","year":"1995"},{"key":"S0022481200004655_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(01)00023-9"},{"key":"S0022481200004655_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0094103"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200004655","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,1]],"date-time":"2019-05-01T00:25:03Z","timestamp":1556670303000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200004655\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,3]]},"references-count":15,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2008,3]]}},"alternative-id":["S0022481200004655"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1208358746","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,3]]}}}