{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T07:31:02Z","timestamp":1772436662704,"version":"3.50.1"},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":1288,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2010,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let<jats:italic>\u03bb<\/jats:italic>denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002528_inline1\"\/>together with 2<jats:sup>\u03bb<\/jats:sup>=<jats:italic>\u03bb<jats:sup>+<\/jats:sup><\/jats:italic>implies \u22c4<jats:sub>S<\/jats:sub>for every<jats:italic>S \u2286 \u03bb<jats:sup>+<\/jats:sup><\/jats:italic>that reflects stationarily often.<\/jats:p><jats:p>In this paper, for a set<jats:italic>S \u2286 \u03bb<jats:sup>+<\/jats:sup><\/jats:italic>, a normal subideal of the weak approachability ideal is introduced, and denoted by<jats:italic>I<\/jats:italic>[<jats:italic>S<\/jats:italic>; \u03bb]. We say that the ideal is<jats:italic>fat<\/jats:italic>if it contains a stationary set. It is proved:<\/jats:p><jats:p>1. if<jats:italic>I<\/jats:italic>[<jats:italic>S<\/jats:italic>; \u03bb] is fat, then NS<jats:sub>\u03bb<\/jats:sub>+ \u2223<jats:italic>S<\/jats:italic>is non-saturated;<\/jats:p><jats:p>2. if<jats:italic>I<\/jats:italic>[<jats:italic>S<\/jats:italic>; \u03bb] is fat and 2<jats:sup>\u03bb<\/jats:sup>= \u03bb<jats:sup>+<\/jats:sup>, then \u22c4<jats:sub>S<\/jats:sub>holds;<\/jats:p><jats:p>3.<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002528_inline1\"\/>implies that<jats:italic>I<\/jats:italic>[<jats:italic>S<\/jats:italic>; \u03bb] is fat for every<jats:italic>S<\/jats:italic>\u2286<jats:italic>\u03bb<\/jats:italic><jats:sup>+<\/jats:sup>that reflects stationarily often;<\/jats:p><jats:p>4. it is relatively consistent with the existence of a supercompact cardinal that<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002528_inline1\"\/>fails, while<jats:italic>I<\/jats:italic>[<jats:italic>S<\/jats:italic>; \u03bb] is fat for every stationary<jats:italic>S \u2286 \u03bb<\/jats:italic><jats:sup>+<\/jats:sup>that reflects stationarily often.<\/jats:p><jats:p>The stronger principle<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002528_inline2\"\/>is studied as well.<\/jats:p>","DOI":"10.2178\/jsl\/1278682214","type":"journal-article","created":{"date-parts":[[2010,7,9]],"date-time":"2010-07-09T13:30:45Z","timestamp":1278682245000},"page":"1035-1065","source":"Crossref","is-referenced-by-count":18,"title":["A relative of the approachability ideal, diamond and 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