{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T22:08:42Z","timestamp":1648850922506},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":1927,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2008,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The game <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004114_inline1\" \/> is played on a complete Boolean algebra <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004114_inline2\" \/>, by two players. White and Black, in <jats:italic>\u03ba<\/jats:italic>-many moves (where <jats:italic>\u03ba<\/jats:italic> is an infinite cardinal). At the beginning White chooses a non-zero element <jats:italic>p<\/jats:italic> \u2208 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004114_inline2\" \/>. In the <jats:italic>\u03b1<\/jats:italic>-th move White chooses <jats:italic>p<\/jats:italic><jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> \u2208 (0, <jats:italic>p<\/jats:italic>)<jats:sub><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004114_inline2\" \/><\/jats:sub> and Black responds choosing <jats:italic>i<\/jats:italic><jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> \u2208{0, 1}. White winsthe play iff <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004114_inline3\" \/>. where <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004114_inline4\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004114_inline5\" \/>.<\/jats:p><jats:p>The corresponding game theoretic properties of c.B.a.'s are investigated. So, Black has a winning strategy (w.s.) if <jats:italic>\u03ba \u2265 \u03c0<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004114_inline2\" \/>) or if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004114_inline2\" \/> contains a <jats:italic>\u03ba<\/jats:italic>-closed dense subset. On the other hand, if White has a w.s., then <jats:italic>\u03ba<\/jats:italic> \u2208 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004114_inline6\" \/>. The existence of w.s. is characterized in a combinatorial way and in terms of forcing. In particular, if 2<jats:sup><jats:italic>&lt;\u03ba<\/jats:italic><\/jats:sup> = <jats:italic>\u03ba<\/jats:italic> \u2208 Reg and forcing by <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004114_inline2\" \/> preserves the regularity of <jats:italic>\u03ba<\/jats:italic>, then White has a w.s. iff the power 2<jats:sup><jats:italic>\u03ba<\/jats:italic><\/jats:sup> is collapsed to <jats:italic>\u03ba<\/jats:italic> in some extension. It is shown that, under the GCH, for each set <jats:italic>S<\/jats:italic> \u2286 Reg there is a c.B.a. <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004114_inline2\" \/> such that White (respectively. Black) has a w.s. for each infinite cardinal <jats:italic>\u03ba<\/jats:italic> \u2208 <jats:italic>S<\/jats:italic> (resp. <jats:italic>\u03ba<\/jats:italic> \u2209 <jats:italic>S<\/jats:italic>). Also it is shown consistent that for each <jats:italic>\u03ba<\/jats:italic> \u2208 Reg there is a c.B.a. on which the game <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004114_inline1\" \/> is undetermined.<\/jats:p>","DOI":"10.2178\/jsl\/1230396930","type":"journal-article","created":{"date-parts":[[2008,12,27]],"date-time":"2008-12-27T16:55:26Z","timestamp":1230396926000},"page":"1433-1457","source":"Crossref","is-referenced-by-count":1,"title":["Power-collapsing games"],"prefix":"10.1017","volume":"73","author":[{"given":"Milo\u0161 S.","family":"Kurili\u0107","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Boris","family":"\u0160obot","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200004114_ref006","unstructured":"Kurili\u0107 M. S. and \u0160obot B. , A game on Boolean algebras describing the collapse of the continuum, submitted."},{"key":"S0022481200004114_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-03-07197-1"},{"key":"S0022481200004114_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(84)90038-1"},{"key":"S0022481200004114_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-02-06501-2"},{"key":"S0022481200004114_ref005","volume-title":"Set theory, An introduction to independence proofs","author":"Kunen","year":"1980"},{"key":"S0022481200004114_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-22400-7"},{"key":"S0022481200004114_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(95)00002-X"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200004114","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,30]],"date-time":"2019-04-30T21:36:00Z","timestamp":1556660160000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200004114\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,12]]},"references-count":7,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2008,12]]}},"alternative-id":["S0022481200004114"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1230396930","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,12]]}}}