{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,8,5]],"date-time":"2022-08-05T15:11:19Z","timestamp":1659712279311},"reference-count":9,"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":4394,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>B<\/jats:italic> be a superatomic Boolean algebra (BA). The rank of <jats:italic>B<\/jats:italic> (rk(<jats:italic>B<\/jats:italic>)). is defined to be the Cantor Bendixon rank of the Stone space of <jats:italic>B<\/jats:italic>. If <jats:italic>a<\/jats:italic> \u2208 <jats:italic>B<\/jats:italic> \u2212 {0}, then the rank of <jats:italic>a<\/jats:italic> in <jats:italic>B<\/jats:italic> (rk(<jats:italic>a<\/jats:italic>)). is defined to be the rank of the Boolean algebra <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200010057_inline1\" \/>. The rank of 0<jats:sup><jats:italic>B<\/jats:italic><\/jats:sup> is defined to be \u22121. An element <jats:italic>a<\/jats:italic> \u2208 <jats:italic>B<\/jats:italic> \u2212 {0} is a generalized atom <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200010057_inline2\" \/>, if the last nonzero cardinal in the cardinal sequence of <jats:italic>B<\/jats:italic> \u21be <jats:italic>a<\/jats:italic> is 1. Let <jats:italic>a, b<\/jats:italic> \u2208 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200010057_inline3\" \/>. We denote <jats:italic>a<\/jats:italic> \u02dc <jats:italic>b<\/jats:italic>, if rk(<jats:italic>a<\/jats:italic>) = rk(<jats:italic>b<\/jats:italic>) = rk(<jats:italic>a<\/jats:italic> \u00b7 <jats:italic>b<\/jats:italic>). A subset <jats:italic>H<\/jats:italic> \u2286 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200010057_inline3\" \/> is a complete set of representatives (CSR) for <jats:italic>B<\/jats:italic>, if for every <jats:italic>a<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200010057_inline3\" \/> there is a unique <jats:italic>h<\/jats:italic> \u2208 <jats:italic>H<\/jats:italic> such that <jats:italic>h<\/jats:italic> ~ <jats:italic>a<\/jats:italic>. Any CSR for <jats:italic>B<\/jats:italic> generates <jats:italic>B<\/jats:italic>. We say that <jats:italic>B<\/jats:italic> is canonically well-generated (CWG), if it has a CSR <jats:italic>H<\/jats:italic> such that the sublattice of <jats:italic>B<\/jats:italic> generated by <jats:italic>H<\/jats:italic> is well-founded. We say that <jats:italic>B<\/jats:italic> is well-generated, if it has a well-founded sublattice <jats:italic>L<\/jats:italic> such that <jats:italic>L<\/jats:italic> generates <jats:italic>B<\/jats:italic>.<\/jats:p><jats:p>Theorem 1. Let <jats:italic>B<\/jats:italic> be a Boolean algebra with cardinal sequence <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200010057_inline4\" \/>. If <jats:italic>B<\/jats:italic> is CWG, then every subalgebra of <jats:italic>B<\/jats:italic> is CWG.<\/jats:p><jats:p>A superatomic Boolean algebra <jats:italic>B<\/jats:italic> is essentially low (ESL), if it has a countable ideal <jats:italic>I<\/jats:italic> such that rk(<jats:italic>B<\/jats:italic>\/<jats:italic>I<\/jats:italic>) \u2264 1.<\/jats:p><jats:p>Theorem 1 follows from Theorem 2.9. which is the main result of this work. For an ESL BA <jats:italic>B<\/jats:italic> we define a set <jats:italic>F<jats:sup>B<\/jats:sup><\/jats:italic> of partial functions from a certain countably infinite set to \u03c9 (Definition 2.8). Theorem 2.9 says that if <jats:italic>B<\/jats:italic> is an ESL Boolean algebra, then the following are equivalent.<\/jats:p><jats:p>(1) Every subalgebra of <jats:italic>B<\/jats:italic> is CWG: and<\/jats:p><jats:p>(2) <jats:italic>F<jats:sup>B<\/jats:sup><\/jats:italic> is bounded.<\/jats:p><jats:p>Theorem 2. If an ESL Boolean algebra is not CWG, then it has a subalgebra which is not well-generated.<\/jats:p>","DOI":"10.2178\/jsl\/1190150050","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T19:12:10Z","timestamp":1197573130000},"page":"369-396","source":"Crossref","is-referenced-by-count":2,"title":["On essentially low, canonically well-generated Boolean algebras"],"prefix":"10.1017","volume":"67","author":[{"given":"Robert","family":"Bonnet","sequence":"first","affiliation":[]},{"given":"Matatyahu","family":"Rubin","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200010057_ref002","unstructured":"Avraham U. , Bonnet R. , Rubin M. , and Si-Kaddour H. , On posets algebras, 43 pages, submitted."},{"key":"S0022481200010057_ref008","volume-title":"Handbook on Boolean Algebras","volume":"1","author":"Koppelberg","year":"1989"},{"key":"S0022481200010057_ref007","first-page":"139","article-title":"Eine Bemerkung zur Theoris der regressiven functionen","volume":"17","author":"Fodor","year":"1965","journal-title":"Acta Scientiarum Mathematicarum"},{"key":"S0022481200010057_ref009","doi-asserted-by":"publisher","DOI":"10.4064\/cm-19-1-59-66"},{"key":"S0022481200010057_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02784128"},{"key":"S0022481200010057_ref006","first-page":"27","article-title":"Skula spaces","volume":"31","author":"Dow","year":"1990","journal-title":"Commentationes Mathematicae Universitatis Carolinae"},{"key":"S0022481200010057_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(99)00034-2"},{"key":"S0022481200010057_ref004","unstructured":"Bonnet R. , A note on well generated Boolean algebras in models satisfying Martin's Axiom, 23 pages, 2000, accepted in Discrete Mathematics ."},{"key":"S0022481200010057_ref005","first-page":"27","volume-title":"Israel Journal of Mathematics","author":"Bonnet"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200010057","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,7]],"date-time":"2019-05-07T01:35:08Z","timestamp":1557192908000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200010057\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,3]]},"references-count":9,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2002,3]]}},"alternative-id":["S0022481200010057"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1190150050","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,3]]}}}