{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,30]],"date-time":"2023-10-30T08:53:19Z","timestamp":1698655999780},"reference-count":6,"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:p>Recall that a subset <jats:italic>X<\/jats:italic> of an algebra <jats:italic>A<\/jats:italic> is <jats:italic>irredundant<\/jats:italic> iff <jats:italic>x<\/jats:italic> \u2209 \u3008<jats:italic>X<\/jats:italic>\u2216{<jats:italic>x<\/jats:italic>}\u3009 for all <jats:italic>x<\/jats:italic> \u03f5 <jats:italic>X<\/jats:italic>, where \u3008<jats:italic>X<\/jats:italic>\u2216{<jats:italic>x<\/jats:italic>}) is the subalgebra generated by <jats:italic>X<\/jats:italic>\u2216{<jats:italic>x<\/jats:italic>}. By Zorn's lemma there is always a maximal irredundant set in an algebra. This gives rise to a natural cardinal function Irr<jats:sub>mm<\/jats:sub>(<jats:italic>A<\/jats:italic>) = min{\u2223<jats:italic>X<\/jats:italic>\u2223: <jats:italic>X<\/jats:italic> is a maximal irredundant subset of <jats:italic>A<\/jats:italic>}. The first half of this article is devoted to proving that there is an atomless Boolean algebra <jats:italic>A<\/jats:italic> of size 2<jats:sup>\u03c9<\/jats:sup> for which Irr<jats:sub>mm<\/jats:sub>(<jats:italic>A<\/jats:italic>) = \u03c9.<\/jats:p><jats:p>A subset <jats:italic>X<\/jats:italic> of a BA <jats:italic>A<\/jats:italic> is <jats:italic>ideal independent<\/jats:italic> iff <jats:italic>x<\/jats:italic> \u2209 (<jats:italic>X<\/jats:italic>\u2216{<jats:italic>x<\/jats:italic>}\u3009<jats:sup>id<\/jats:sup> for all <jats:italic>x<\/jats:italic> \u03f5 <jats:italic>X<\/jats:italic>, where \u3008<jats:italic>X<\/jats:italic>\u2216{<jats:italic>x<\/jats:italic>}\u3009<jats:sup>id<\/jats:sup> is the ideal generated by <jats:italic>X<\/jats:italic>\u2216{<jats:italic>x<\/jats:italic>}. Again, by Zorn's lemma there is always a maximal ideal independent subset of any Boolean algebra. We then consider two associated functions. A spectrum function<\/jats:p><jats:p>S<jats:sub>spect<\/jats:sub>(<jats:italic>A<\/jats:italic>) = {\u2223<jats:italic>X<\/jats:italic>\u2223: <jats:italic>X<\/jats:italic> is a maximal ideal independent subset of <jats:italic>A<\/jats:italic>}<\/jats:p><jats:p>and the least element of this set, s<jats:sub>mm<\/jats:sub>(<jats:italic>A<\/jats:italic>). We show that many sets of infinite cardinals can appear as S<jats:sub>spect<\/jats:sub>(<jats:italic>A<\/jats:italic>). The relationship of S<jats:sub>mm<\/jats:sub> to similar \u201ccontinuum cardinals\u201d is investigated. It is shown that it is relatively consistent that S<jats:sub>mm<\/jats:sub><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004722_inline1\" \/>\/fin) &lt; 2<jats:sup>\u03c9<\/jats:sup>.<\/jats:p><jats:p>We use the letter <jats:italic>s<\/jats:italic> here because of the relationship of ideal independence with the well-known cardinal invariant <jats:italic>spread<\/jats:italic>; see Monk [5]. Namely, sup{\u2223<jats:italic>X<\/jats:italic>\u2223: <jats:italic>X<\/jats:italic> is ideal independent in <jats:italic>A<\/jats:italic>} is the same as the spread of the Stone space Ult(<jats:italic>A<\/jats:italic>); the spread of a topological space <jats:italic>X<\/jats:italic> is the supremum of cardinalities of discrete subspaces.<\/jats:p>","DOI":"10.2178\/jsl\/1208358753","type":"journal-article","created":{"date-parts":[[2008,10,14]],"date-time":"2008-10-14T15:19:02Z","timestamp":1223997542000},"page":"261-275","source":"Crossref","is-referenced-by-count":2,"title":["Maximal irredundance and maximal ideal independence in Boolean algebras"],"prefix":"10.1017","volume":"73","author":[{"given":"J. Donald","family":"Monk","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200004722_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0346-0334-8"},{"key":"S0022481200004722_ref003","volume-title":"Set theory","author":"Kunen","year":"1980"},{"key":"S0022481200004722_ref002","volume-title":"Handbook on Boolean algebras","volume":"1","author":"Koppelberg","year":"1989"},{"key":"S0022481200004722_ref001","volume-title":"Handbook of set theory","author":"Blass"},{"key":"S0022481200004722_ref006","first-page":"1928","volume":"66","author":"Monk","year":"2001","journal-title":"Continuum cardinals generalized to Boolean algebras"},{"key":"S0022481200004722_ref004","first-page":"674","volume":"69","author":"Mckenzie","year":"2004","journal-title":"On some small cardinals for Boolean algebras"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200004722","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,1]],"date-time":"2019-05-01T00:24:40Z","timestamp":1556670280000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200004722\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,3]]},"references-count":6,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2008,3]]}},"alternative-id":["S0022481200004722"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1208358753","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,3]]}}}