{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T07:59:24Z","timestamp":1775462364892,"version":"3.50.1"},"reference-count":6,"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":9598,"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":[[1987,12]]},"abstract":"Une cons\u00e9quence de la classification des th\u00e9ories compl\u00e8tes d'alg\u00e8bres de Boole par Tarski [5] est que la th\u00e9orie \u00e9l\u00e9mentaire d'une alg\u00e8bre de Boole A<\/jats:italic> est d\u00e9termin\u00e9e par le type d'isomorphisme du treillis de ses id\u00e9aux d\u00e9finissables et, pour chacun de ces id\u00e9aux, par le nombre d'atomes du quotient de A<\/jats:italic> par cet id\u00e9al lorsque ce nombre est fini. Une remarque analogue peut \u00eatre faite \u00e0 propos des cas particuliers d'alg\u00e8bres de Boole munies d'un id\u00e9al distingu\u00e9 \u00e9tudi\u00e9s par Ershov [1] et par Jurie et Touraille [3]; dans to us ces cas, c'est la simplicit\u00e9 des treillis possibles qui permet la classification des th\u00e9ories compl\u00e8tes. Le r\u00e9sultat principal de cet article est que, dans le cas g\u00e9n\u00e9ral d'une alg\u00e8bre de Boole munie d'une famille quelconque d'id\u00e9aux distingu\u00e9s, la th\u00e9orie d'un mod\u00e8le peut encore \u00eatre caract\u00e9ris\u00e9e gr\u00e2ce \u00e0 une structure alg\u00e9brique sur l'ensemble de ses id\u00e9aux d\u00e9finissables. Il s'agit d'une structure d'alg\u00e8bre de Heyting munie d'une op\u00e9ration unaire sa d\u00e9finie par sa(K<\/jats:italic>) = {a<\/jats:italic>: a<\/jats:italic>\/K<\/jats:italic> est sans atome}, et cette structure s'av\u00e8re \u00eatre engendr\u00e9e par les id\u00e9aux distingu\u00e9s du mod\u00e8le. La m\u00e9thode utilis\u00e9e est l'\u00e9limination directe des quantificateurs, par r\u00e9ductions successives des formules. Elle n\u00e9cessite des propri\u00e9t\u00e9s alg\u00e9briques et topologiques qui sont donn\u00e9es aux \u00a7\u00a71 et 2: on introduit au \u00a71 la notion d'alg\u00e8bre de Heyting \u00e9toil\u00e9e, c'est-\u00e0-dire d'alg\u00e8bre de Heyting munie d'une op\u00e9ration unaire * v\u00e9rifiant des \u00e9galit\u00e9s qui permettent de rendre compte, d'une certaine fa\u00e7on, de la d\u00e9rivation de Cantor-Bendixon; le \u00a72 est consacr\u00e9 \u00e0 des propri\u00e9t\u00e9s topologiques qui, dans le cas de l'espace de Stone d'une alg\u00e8bre de Boole A<\/jats:italic>, permettent d'\u00e9claircir les relations possibles entre les atomes des quotients de A<\/jats:italic> par des id\u00e9aux diff\u00e9rents.<\/jats:p>","DOI":"10.2307\/2273836","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:23:41Z","timestamp":1146939821000},"page":"1027-1043","source":"Crossref","is-referenced-by-count":11,"title":["Th\u00e9ories d'alg\u00e8bres de Boole munies d'id\u00e9aux distingu\u00e9s. I: Th\u00e9ories \u00e9l\u00e9mentaires"],"prefix":"10.1017","volume":"52","author":[{"given":"Alain","family":"Touraille","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200029418_ref005","first-page":"64","article-title":"Arithmetical classes and types of Boolean algebras","volume":"55","author":"Tarski","year":"1949","journal-title":"Bulletin of the American Mathematical Society"},{"key":"S0022481200029418_ref004","doi-asserted-by":"publisher","DOI":"10.2307\/1969038"},{"key":"S0022481200029418_ref003","first-page":"415","article-title":"Id\u00e9aux \u00e9l\u00e9mentairement \u00e9quivalents dans une alg\u00e8bre bool\u00e9ienne","volume":"299","author":"Jurie","year":"1984","journal-title":"Comptes Rendus des Seances de l'Acad\u00e9mie des Sciences. S\u00e9rie I: Math\u00e9matique"},{"key":"S0022481200029418_ref002","doi-asserted-by":"publisher","DOI":"10.4064\/fm-47-1-57-103"},{"key":"S0022481200029418_ref001","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":"S0022481200029418_ref006","first-page":"125","article-title":"\u00c9limination des quantificateurs dans la th\u00e9orie \u00e9l\u00e9mentaire des alg\u00e8bres de Boole munies d'une famille d'id\u00e9aux distingu\u00e9s","volume":"300","author":"Touraille","year":"1985","journal-title":"Comptes Rendus des Seances de l'Acad\u00e9mie des Sciences. S\u00e9rie I: Math\u00e9matique"}],"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\/S0022481200029418","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,20]],"date-time":"2019-05-20T17:16:59Z","timestamp":1558372619000},"score":1,"resource":{"primary":{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.cambridge.org\/core\/product\/identifier\/S0022481200029418\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1987,12]]},"references-count":6,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1987,12]]}},"alternative-id":["S0022481200029418"],"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.2307\/2273836","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1987,12]]}}}