{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T23:52:17Z","timestamp":1649029937770},"reference-count":6,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":9781,"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1987,6]]},"abstract":"<jats:p>Throughout this paper, <jats:italic>B<\/jats:italic> will always be a Boolean algebra and <jats:italic>\u0393<\/jats:italic> an ultrafilter on <jats:italic>B<\/jats:italic>. We use + and <jats:italic>\u03a3<\/jats:italic> for the Boolean join operation and \u00b7 and \u03a0 for the Boolean meet.<\/jats:p><jats:p><jats:italic>\u03ba<\/jats:italic> is always a regular cardinal. <jats:italic>C<\/jats:italic>(<jats:italic>\u03ba<\/jats:italic>) is the full structure of <jats:italic>\u03ba<\/jats:italic>, the structure with universe <jats:italic>\u03ba<\/jats:italic> and whose functions and relations consist of all unitary functions and relations on <jats:italic>\u03ba<\/jats:italic>. <jats:italic>\u03ba<\/jats:italic><jats:sup><jats:italic>B<\/jats:italic><\/jats:sup> is the collection of all <jats:italic>B<\/jats:italic>-valued names for elements of <jats:italic>\u03ba<\/jats:italic>. We use symbols <jats:italic>f, g, h<\/jats:italic> for members of <jats:italic>\u03ba<\/jats:italic><jats:sup><jats:italic>B<\/jats:italic><\/jats:sup>. Formally an element <jats:italic>f<\/jats:italic> \u2208 <jats:italic>\u03ba<\/jats:italic><jats:sup><jats:italic>B<\/jats:italic><\/jats:sup> is a mapping <jats:italic>\u03ba<\/jats:italic> \u2192 <jats:italic>B<\/jats:italic> with the properties that <jats:italic>\u03a3<\/jats:italic><jats:sub><jats:italic>\u03b1<\/jats:italic>\u2208<jats:italic>\u03ba<\/jats:italic><\/jats:sub><jats:italic>f<\/jats:italic>(<jats:italic>\u03b1<\/jats:italic>) = 1<jats:sub><jats:italic>B<\/jats:italic><\/jats:sub> and that <jats:italic>f<\/jats:italic>(<jats:italic>\u03b1<\/jats:italic>) \u00b7 <jats:italic>f<\/jats:italic>(<jats:italic>\u03b2<\/jats:italic>) = 0<jats:sub><jats:italic>B<\/jats:italic><\/jats:sub> whenever <jats:italic>\u03b1<\/jats:italic> \u2260 <jats:italic>\u03b2<\/jats:italic>. We view <jats:italic>f<\/jats:italic>(<jats:italic>\u03b1<\/jats:italic>) as the Boolean-truth value indicating the extent to which the name <jats:italic>f<\/jats:italic> is equal to <jats:italic>\u03b1<\/jats:italic>, and we will hereafter write \u2225<jats:italic>f<\/jats:italic> = <jats:italic>\u03b1<\/jats:italic>\u2225 for <jats:italic>f<\/jats:italic>(<jats:italic>\u03b1<\/jats:italic>). For every <jats:italic>\u03b1<\/jats:italic> \u2208 <jats:italic>\u03ba<\/jats:italic> there is a canonical name <jats:italic>f<\/jats:italic><jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> \u2208 <jats:italic>\u03ba<\/jats:italic><jats:sup><jats:italic>B<\/jats:italic><\/jats:sup> which has the property that \u2225<jats:italic>f<\/jats:italic><jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> = <jats:italic>\u03b1<\/jats:italic>\u2225 = 1. Hereafter we identify <jats:italic>\u03b1<\/jats:italic> and <jats:italic>f<\/jats:italic><jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub>.<\/jats:p><jats:p>If <jats:italic>B<\/jats:italic> is a <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>-complete Boolean algebra and <jats:italic>\u0393<\/jats:italic> is an ultrafilter on <jats:italic>B<\/jats:italic>, then we may define the Boolean ultraproduct <jats:italic>C<\/jats:italic>(<jats:italic>\u03ba<\/jats:italic>)<jats:sup><jats:italic>B<\/jats:italic><\/jats:sup>\/<jats:italic>\u0393<\/jats:italic> in the following manner. If <jats:italic>\u03d5<\/jats:italic>(<jats:italic>x<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>x<jats:sub>n<\/jats:sub><\/jats:italic>) is a formula of <jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03ba<\/jats:italic><\/jats:sub>, the language for <jats:italic>C<\/jats:italic>(<jats:italic>\u03ba<\/jats:italic>) (which has symbols for all finitary functions and relations on <jats:italic>\u03ba<\/jats:italic>), and <jats:italic>f<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>f<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>f<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u22121<\/jats:sub> are elements of <jats:italic>\u03ba<\/jats:italic><jats:sup><jats:italic>B<\/jats:italic><\/jats:sup> then we define the Boolean-truth value of <jats:italic>\u03d5<\/jats:italic>(<jats:italic>f<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>f<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>f<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u22121<\/jats:sub>) as<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200034460_eqnU1\" \/><\/jats:disp-formula><\/jats:p>","DOI":"10.2307\/2274400","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:21:40Z","timestamp":1146954100000},"page":"530-542","source":"Crossref","is-referenced-by-count":0,"title":["Complete Boolean ultraproducts"],"prefix":"10.1017","volume":"52","author":[{"given":"R. Michael","family":"Canjar","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200034460_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90017-9"},{"key":"S0022481200034460_ref004","volume-title":"Set theory: an introduction to independence proofs","author":"Kunen","year":"1980"},{"key":"S0022481200034460_ref003","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S0022481200034460_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-65780-1"},{"key":"S0022481200034460_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71561-1"},{"key":"S0022481200034460_ref001","unstructured":"Blass A. , Orderings of ultrafilters, Ph.D. Thesis, Harvard University, Cambridge, Massachusetts, 1970."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200034460","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,21]],"date-time":"2019-05-21T19:47:59Z","timestamp":1558468079000},"score":1,"resource":{"primary":{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.cambridge.org\/core\/product\/identifier\/S0022481200034460\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1987,6]]},"references-count":6,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1987,6]]}},"alternative-id":["S0022481200034460"],"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.2307\/2274400","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1987,6]]}}}