{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T08:16:48Z","timestamp":1648541808992},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":8137,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1991,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>For a complete theory of Boolean algebras <jats:italic>T<\/jats:italic>, let <jats:italic>M<jats:sub>T<\/jats:sub><\/jats:italic> denote the class of countable models of <jats:italic>T<\/jats:italic>. For <jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>B<\/jats:italic><jats:sub>2<\/jats:sub> \u2208 <jats:italic>M<jats:sub>T<\/jats:sub><\/jats:italic>, let <jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub> \u2264 <jats:italic>B<\/jats:italic><jats:sub>2<\/jats:sub> mean that <jats:italic>B<\/jats:italic><jats:sub>2<\/jats:sub> is elementarily embeddable in <jats:italic>B<\/jats:italic><jats:sub>2<\/jats:sub>. Theorem 1. <jats:italic>For every complete theory of Boolean algebras T<\/jats:italic>, <jats:italic>if T<\/jats:italic> \u2260 <jats:italic>T<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub>, <jats:italic>then<\/jats:italic> \u2039<jats:italic>M<jats:sub>T<\/jats:sub><\/jats:italic>, \u2264\u203a <jats:italic>is well-quasi-ordered<\/jats:italic>. \u220e We define <jats:italic>T<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub>. For a Boolean algebra <jats:italic>B<\/jats:italic>, let <jats:italic>I(B)<\/jats:italic> be the ideal of all elements of the form <jats:italic>a<\/jats:italic> + <jats:italic>s<\/jats:italic> such that <jats:italic>B<\/jats:italic> \u21be <jats:italic>a<\/jats:italic> is an atomic Boolean algebra and <jats:italic>B<\/jats:italic> \u21be <jats:italic>s<\/jats:italic> is an atomless Boolean algebra. The <jats:italic>Tarski derivative<\/jats:italic> of <jats:italic>B<\/jats:italic> is defined as follows: <jats:italic>B<\/jats:italic><jats:sup>(0)<\/jats:sup> = <jats:italic>B<\/jats:italic> and <jats:italic>B<\/jats:italic><jats:sup>(<jats:italic>n<\/jats:italic> + 1)<\/jats:sup> = <jats:italic>B<jats:sup>(n)<\/jats:sup><\/jats:italic>\/<jats:italic>I<\/jats:italic>(<jats:italic>B<jats:sup>(n)<\/jats:sup><\/jats:italic>). Define <jats:italic>T<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub> to be the theory of all Boolean algebras such that for every <jats:italic>n<\/jats:italic> \u2208 <jats:italic>\u03c9<\/jats:italic>, <jats:italic>B<jats:sup>(n)<\/jats:sup><\/jats:italic> \u2260 {0}. By Tarski [1949], <jats:italic>T<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub> is complete. Recall that \u2039<jats:italic>A<\/jats:italic>, &lt; \u203a is <jats:italic>partial well-quasi-ordering<\/jats:italic>, it is a partial quasi-ordering and for every {<jats:italic>a<jats:sub>i<\/jats:sub><\/jats:italic>, \u20d2 <jats:italic>i<\/jats:italic> \u2208 <jats:italic>\u03c9<\/jats:italic>} \u2286 <jats:italic>A<\/jats:italic>, there are <jats:italic>i<\/jats:italic> &lt; <jats:italic>j<\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic> such that <jats:italic>a<jats:sub>i<\/jats:sub><\/jats:italic> \u2264 <jats:italic>a<jats:sub>j<\/jats:sub><\/jats:italic>. Theorem 2. <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200023550_inline1\" \/><jats:italic>contains a subset M such that the partial orderings<\/jats:italic> \u2039<jats:italic>M<\/jats:italic>, \u2264 \u21be <jats:italic>M<\/jats:italic>\u203a and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200023550_inline2\" \/><jats:italic>are isomorphic<\/jats:italic>. \u220e Let <jats:italic>M<\/jats:italic>\u2032<jats:sub>0<\/jats:sub> denote the class of all countable Boolean algebras. For <jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>B<\/jats:italic><jats:sub>2<\/jats:sub> \u2208 <jats:italic>M<\/jats:italic>\u2032<jats:sub>0<\/jats:sub>, let <jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub> \u2264\u2032 <jats:italic>B<\/jats:italic><jats:sub>2<\/jats:sub> mean that <jats:italic>B<\/jats:italic><jats:sub>1<\/jats:sub> is embeddable in <jats:italic>B<\/jats:italic><jats:sub>2<\/jats:sub>. Remark. \u2039<jats:italic>M<\/jats:italic>\u2032<jats:sub>0<\/jats:sub>, \u2264\u2032\u203a <jats:italic>is well-quasi-ordered<\/jats:italic>. \u220e This follows from Laver's theorem [1971] that the class of countable linear orderings with the embeddability relation is well-quasi-ordered and the fact that every countable Boolean algebra is isomorphic to a Boolean algebra of a linear ordering.<\/jats:p>","DOI":"10.2307\/2275469","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:42:28Z","timestamp":1146940948000},"page":"1212-1229","source":"Crossref","is-referenced-by-count":0,"title":["Elementary embedding between countable Boolean algebras"],"prefix":"10.1017","volume":"56","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":"S0022481200023550_ref008","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100039062"},{"key":"S0022481200023550_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02483113"},{"key":"S0022481200023550_ref012","first-page":"1192","article-title":"Arithmetical classes and types of Boolean algebras: preliminary report","volume":"55","author":"Tarski","year":"1949","journal-title":"Bulletin of the American Mathematical Society"},{"key":"S0022481200023550_ref004","volume-title":"Theory of relations","author":"Fra\u00efss\u00e9","year":"1986"},{"key":"S0022481200023550_ref002","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200023550_ref003","first-page":"17","article-title":"Decidability of the elementary theory of relatively complemented distributive lattices and the theory of filters","volume":"3","author":"Ershov","year":"1964","journal-title":"Algebra i Logika Seminar"},{"key":"S0022481200023550_ref005","volume-title":"Handbook of Boolean algebras","volume":"1","author":"Koppelberg","year":"1989"},{"key":"S0022481200023550_ref006","doi-asserted-by":"publisher","DOI":"10.2307\/1970754"},{"key":"S0022481200023550_ref007","first-page":"487","volume-title":"Graphs and order (Banff, 1984","volume":"147","author":"Milner","year":"1984"},{"key":"S0022481200023550_ref009","doi-asserted-by":"publisher","DOI":"10.4064\/cm-23-1-5-15"},{"key":"S0022481200023550_ref011","first-page":"503","volume-title":"Graphs and order (Banff, 1984","volume":"147","author":"Pouzet","year":"1984"},{"key":"S0022481200023550_ref010","doi-asserted-by":"publisher","DOI":"10.4064\/fm-78-1-43-60"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200023550","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T16:00:32Z","timestamp":1558108832000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200023550\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1991,12]]},"references-count":12,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1991,12]]}},"alternative-id":["S0022481200023550"],"URL":"https:\/\/doi.org\/10.2307\/2275469","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1991,12]]}}}