{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T06:47:35Z","timestamp":1769237255141,"version":"3.49.0"},"reference-count":16,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,11,13]],"date-time":"2006-11-13T00:00:00Z","timestamp":1163376000000},"content-version":"vor","delay-in-days":3238,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[1998,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this article we investigate the dual\u2010shattering cardinal \u210c, the dual\u2010splitting cardinal \ud835\udd16 and the dual\u2010reaping cardinal \ud835\udd0e, which are dualizations of the well\u2010known cardinals \ud835\udd25 (the shattering cardinal, also known as the distributivity number of <jats:italic>P<\/jats:italic>(\u03c9)\/fin), <jats:italic>s<\/jats:italic> (the splitting number) and \ud835\udd20 (the reaping number). Using some properties of the ideal \ud835\udd0d of nowhere dual\u2010Ramsey sets, which is an ideal over the set of partitions of \u03c9, we show that add(\ud835\udd0d) = cov(\ud835\udd0d) = \u210c. With this result we can show that \u210c &gt; \u03c9<jats:sub>1<\/jats:sub> is consistent with ZFC and as a corollary we get the relative consistency of \u210c &gt; \ud835\udd31 <jats:italic>t<\/jats:italic>, where t is the tower number. Concerning \ud835\udd16 we show that cov(<jats:italic>M<\/jats:italic>) \u2a7d \ud835\udd16 \ud835\udd2f (where <jats:italic>M<\/jats:italic> is the ideal of the meager sets). For the dual\u2010reaping cardinal \ud835\udd16 we get <jats:italic>p<\/jats:italic> \ud835\udd16 \u2a7d \ud835\udd2d \u2a7d \ud835\udd2f (where \ud835\udd2d is the pseudo\u2010intersection number) and for a modified dual\u2010reaping number \ud835\udd16\u2032 we get \ud835\udd16\u2032 \u2a7d \ud835\udd2c (where \ud835\udd2c is the dominating number). As a consistency result we get \ud835\udd16 &lt; cov(\ud835\udd10).<\/jats:p>","DOI":"10.1002\/malq.19980440109","type":"journal-article","created":{"date-parts":[[2007,5,31]],"date-time":"2007-05-31T02:47:57Z","timestamp":1180579677000},"page":"123-134","source":"Crossref","is-referenced-by-count":9,"title":["On Shattering, Splitting and Reaping Partitions"],"prefix":"10.1002","volume":"44","author":[{"given":"Lorenz","family":"Halbeisen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,11,13]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"crossref","first-page":"11","DOI":"10.4064\/fm-110-1-11-24","article-title":"The space of ultrafilters on N covered by nowhere dense sets","volume":"110","author":"Balcar B.","year":"1980","journal-title":"Fund. 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