{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T15:45:32Z","timestamp":1771947932732,"version":"3.50.1"},"reference-count":17,"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":9507,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1988,3]]},"abstract":"<jats:p>Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200029005_inline1.png\"\/> be a complete Boolean algebra and <jats:italic>G<\/jats:italic> a finite simple group in the Scott-Solovay <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200029005_inline1.png\"\/>-valued model <jats:italic>V<\/jats:italic><jats:sup>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200029005_inline1.png\"\/>)<\/jats:sup> of set theory. If we observe <jats:italic>G<\/jats:italic> outside <jats:italic>V<\/jats:italic><jats:sup>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200029005_inline1.png\"\/>)<\/jats:sup>, then we get a new group which is denoted by <jats:italic>\u011c<\/jats:italic>. In general, <jats:italic>\u011c<\/jats:italic> is not finite nor simple. Nevertheless \u011c satisfies every property satisfied by a finite simple group with some translation. In this way, we can get a class of groups for which we can use a well-developed theory of the finite simple groups. We call <jats:italic>\u011c<\/jats:italic> Boolean <jats:italic>simple<\/jats:italic> if <jats:italic>G<\/jats:italic> is simple in some <jats:italic>V<\/jats:italic><jats:sup>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200029005_inline1.png\"\/>)<\/jats:sup>. In the same way we define Boolean simple rings. The main purpose of this paper is a study of structures of Boolean simple groups and Boolean simple rings. As for Boolean simple rings, K. Eda previously constructed Boolean completion of rings with a certain condition. His construction is useful for our purpose.<\/jats:p><jats:p>The present work is a part of a series of systematic applications of Boolean valued method. The reader who is interested in this subject should consult with papers by Eda, Nishimura, Ozawa, and the author in the list of references.<\/jats:p>","DOI":"10.2307\/2274435","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:24:27Z","timestamp":1146954267000},"page":"160-173","source":"Crossref","is-referenced-by-count":5,"title":["Boolean simple groups and boolean simple rings"],"prefix":"10.1017","volume":"53","author":[{"given":"Gaisi","family":"Takeuti","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200029005_bib011","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-32.1.141"},{"key":"S0022481200029005_bib016","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/03510001"},{"key":"S0022481200029005_bib006","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90017-9"},{"key":"S0022481200029005_bib003","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(83)90174-6"},{"key":"S0022481200029005_bib009","first-page":"368","article-title":"Boolean valued analysis and type I AW*-algebras","volume":"59","author":"Ozawa","year":"1983","journal-title":"Proceedings of the Japan Academy"},{"key":"S0022481200029005_bib005","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/03830493"},{"key":"S0022481200029005_bib008","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/03540609"},{"key":"S0022481200029005_bib012","volume-title":"Two applications of logic to mathematics","author":"Takeuti","year":"1978"},{"key":"S0022481200029005_bib015","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0092890"},{"key":"S0022481200029005_bib013","first-page":"417","volume":"44","author":"Takeuti","year":"1979","journal-title":"A transfer principle in harmonic analysis"},{"key":"S0022481200029005_bib002","doi-asserted-by":"publisher","DOI":"10.21099\/tkbjm\/1496159530"},{"key":"S0022481200029005_bib014","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0061842"},{"key":"S0022481200029005_bib001","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/03640619"},{"key":"S0022481200029005_bib004","volume-title":"Algebra","author":"Lang","year":"1971"},{"key":"S0022481200029005_bib007","doi-asserted-by":"publisher","DOI":"10.2977\/prims\/1195179842"},{"key":"S0022481200029005_bib017","doi-asserted-by":"publisher","DOI":"10.4099\/math1924.9.207"},{"key":"S0022481200029005_bib010","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/03640589"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200029005","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T10:38:37Z","timestamp":1679481517000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200029005\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1988,3]]},"references-count":17,"aliases":["10.1017\/s0022481200029005"],"journal-issue":{"issue":"1","published-print":{"date-parts":[[1988,3]]}},"alternative-id":["S0022481200029005"],"URL":"https:\/\/doi.org\/10.2307\/2274435","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1988,3]]}}}