{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T12:00:35Z","timestamp":1759147235971},"reference-count":26,"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>We characterize the (<jats:italic>\u03ba, \u03bb<\/jats:italic>, &lt; <jats:italic>\u03bc<\/jats:italic>)-distributive law in Boolean algebras in terms of cut and choose games <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004710_inline2\" \/>, when <jats:italic>\u03bc<\/jats:italic> \u2264 <jats:italic>\u03ba<\/jats:italic> \u2264 <jats:italic>\u03bb<\/jats:italic> and <jats:italic>\u03ba<\/jats:italic><jats:sup>&lt;<jats:italic>\u03ba<\/jats:italic><\/jats:sup> = <jats:italic>\u03ba<\/jats:italic>. This builds on previous work to yield game-theoretic characterizations of distributive laws for almost all triples of cardinals <jats:italic>\u03ba, \u03bb, \u03bc<\/jats:italic> with <jats:italic>\u03bc<\/jats:italic> \u2264 <jats:italic>\u03bb<\/jats:italic>, under GCH. In the case when <jats:italic>\u03bc<\/jats:italic> \u2264 <jats:italic>\u03ba<\/jats:italic> \u2264 <jats:italic>\u03bb<\/jats:italic> and <jats:italic>\u03ba<\/jats:italic><jats:sup>&lt;<jats:italic>\u03ba<\/jats:italic><\/jats:sup> = <jats:italic>\u03ba<\/jats:italic>, we show that it is necessary to consider whether the <jats:italic>\u03ba<\/jats:italic>-stationarity of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004710_inline1\" \/> in the ground model is preserved by <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004710_inline5\" \/>. In this vein, we develop the theory of <jats:italic>\u03ba<\/jats:italic>-club and <jats:italic>\u03ba<\/jats:italic>-stationary subsets of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004710_inline1\" \/>. We also construct Boolean algebras in which Player I wins <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004710_inline4\" \/> but the (<jats:italic>\u03ba<\/jats:italic>, \u221e, <jats:italic>\u03ba<\/jats:italic>)-d.1. holds, and, assuming GCH, construct Boolean algebras in which many games are undetermined.<\/jats:p>","DOI":"10.2178\/jsl\/1208358752","type":"journal-article","created":{"date-parts":[[2008,10,14]],"date-time":"2008-10-14T11:19:02Z","timestamp":1223983142000},"page":"238-260","source":"Crossref","is-referenced-by-count":3,"title":["<i>\u03ba<\/i>-stationary subsets of , infinitary games, and distributive laws in Boolean 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