{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,26]],"date-time":"2023-10-26T15:12:40Z","timestamp":1698333160528},"reference-count":15,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2006,11,13]],"date-time":"2006-11-13T00:00:00Z","timestamp":1163376000000},"content-version":"vor","delay-in-days":4334,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[1995,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider the theory Th<jats:sub>prin<\/jats:sub> of Boolean algebras with a principal ideal, the theory Th<jats:sub>max<\/jats:sub> of Boolean algebras with a maximal ideal, the theory Th<jats:sub>ac<\/jats:sub> of atomic Boolean algebras with an ideal where the supremum of the ideal exists, and the theory Th<jats:sub>sa<\/jats:sub> of atomless Boolean algebras with an ideal where the supremum of the ideal exists. First, we find elementary invariants for Th<jats:sub>prin<\/jats:sub> and Th<jats:sub>sa.<\/jats:sub> If <jats:italic>T<\/jats:italic> is a theory in a first order language and \u03b1 is a linear order with least element, then we let Sentalg(<jats:italic>T<\/jats:italic>) be the Lindenbaum\u2010Tarski algebra with respect to <jats:italic>T<\/jats:italic>, and we let intalg(\u03b1) be the interval algebra of \u03b1. Using rank diagrams, we show that Sentalg(Th<jats:sub>prin<\/jats:sub>) \u22cd intalg(\u03c9<jats:sup>4<\/jats:sup>), Sentalg(Th<jats:sub>max<\/jats:sub>) \u22cd intalg(\u03c9<jats:sup>3<\/jats:sup>) \u22cd Sentalg(Th<jats:sub>ac<\/jats:sub>), and Sentalg(Th<jats:sub>sa<\/jats:sub>) \u22cd intalg(\u03c9<jats:sup>2<\/jats:sup> + \u03c9<jats:sup>2<\/jats:sup>). For Th<jats:sub>max<\/jats:sub> and Th<jats:sub>ac<\/jats:sub> we use Ershov's elementary invariants of these theories. We also show that the algebra of formulas of the theory <jats:italic>Tx<\/jats:italic> of Boolean algebras with finitely many ideals is atomic.<\/jats:p>","DOI":"10.1002\/malq.19950410406","type":"journal-article","created":{"date-parts":[[2007,6,2]],"date-time":"2007-06-02T20:51:29Z","timestamp":1180817489000},"page":"485-504","source":"Crossref","is-referenced-by-count":1,"title":["Some Boolean Algebras with Finitely Many Distinguished Ideals I"],"prefix":"10.1002","volume":"41","author":[{"given":"Regina","family":"Arag\u00f3n","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","volume-title":"Model Theory","author":"Chang C. C.","year":"1973"},{"key":"e_1_2_1_3_2","first-page":"17","article-title":"Decidability of the elementary theory of relatively complemented lattices and the theory of filters","volume":"3","author":"Ershov Ju. 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Paris"},{"key":"e_1_2_1_7_2","volume-title":"Handbook of Boolean Algebras","author":"Koppelberg S.","year":"1989"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(76)90148-4"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9452-5"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01980754"},{"key":"e_1_2_1_10_3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01980754"},{"key":"e_1_2_1_11_2","first-page":"370","article-title":"Rank diagrams and Boolean algebras","volume":"43","author":"Myers D.","year":"1978","journal-title":"J. Symbolic Logic"},{"key":"e_1_2_1_12_2","first-page":"1167","volume-title":"Handbook of Boolean Algebras","author":"Myers D.","year":"1989"},{"key":"e_1_2_1_13_2","first-page":"1153","article-title":"Lindenbaum\u2010Tarski algebras of Boolean algebras with ideals. Heyting algebras and commutative rings","volume":"58","author":"Pal'chunov D.","year":"1993","journal-title":"J. Symbolic Logic"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.2307\/2273836"},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.2307\/2274482"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.19950410406","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.19950410406","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T16:27:40Z","timestamp":1698251260000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.19950410406"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,1]]},"references-count":15,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1995,1]]}},"alternative-id":["10.1002\/malq.19950410406"],"URL":"https:\/\/doi.org\/10.1002\/malq.19950410406","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,1]]}}}