{"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":1775462364893,"version":"3.50.1"},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":8593,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,9]]},"abstract":"<jats:p>La premi\u00e8re partie de cet article ([10], que nous d\u00e9signerons par (I) dans la suite) \u00e9tait consacr\u00e9e \u00e0 la th\u00e9orie \u00e9l\u00e9mentaire <jats:italic>T<jats:sub>X<\/jats:sub><\/jats:italic> des alg\u00e8bres de Boole munies d'id\u00e9aux distingu\u00e9s ind\u00e9x\u00e9s dans un ensemble <jats:italic>X<\/jats:italic>. On a vu que les id\u00e9aux d\u00e9finissables dans un mod\u00e8le <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025640_inline1\"\/> de <jats:italic>T<jats:sub>x<\/jats:sub><\/jats:italic> forment la sous-alg\u00e8bre engendr\u00e9e par les id\u00e9aux distingu\u00e9s de l'alg\u00e8bre de Heyting des id\u00e9aux de <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025640_inline1\"\/> munie de l'op\u00e9rateur sa d\u00e9fini par sa(<jats:italic>K<\/jats:italic>) = {<jats:italic>a: a\/K<\/jats:italic> est sans atome}, et que la th\u00e9orie de <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025640_inline1\"\/> peut \u00eatre caract\u00e9ris\u00e9e par la structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025640_inline2\"\/> compos\u00e9e de l'alg\u00e8bre de Heyting des id\u00e9aux d\u00e9finissables munie de l'op\u00e9rateur sa et des id\u00e9aux distingu\u00e9s, et par l'application qui \u00e0 tout <jats:italic>K<\/jats:italic> de <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025640_inline2\"\/> fait correspondre le nombre d'atomes de <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025640_inline1\"\/>\/<jats:italic>K<\/jats:italic>, pris dans <jats:bold>N<\/jats:bold> \u22c3 {\u221e}.<\/jats:p><jats:p>Nous montrons maintenant que les structures <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025640_inline2\"\/> possibles peuvent \u00eatre d\u00e9finies de fa\u00e7on axiomatique en introduisant une classe \u00e9quationnelle d'alg\u00e8bres de Heyting munies d'une op\u00e9ration unaire, dites sa-alg\u00e8bres de Heyting (en abr\u00e9g\u00e9 sa-AH), et en prouvant que cette classe est constitu\u00e9e des alg\u00e8bres pouvant \u00eatre plong\u00e9es dans l'alg\u00e8bre de Heyting des id\u00e9aux d'une alg\u00e8bre de Boole, munie de l'op\u00e9rateur sa. Ainsi les <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025640_inline2\"\/> sont, \u00e0 isomorphisme pr\u00e8s, les sa-AH engendr\u00e9es par des \u00e9l\u00e9ments distingu\u00e9s ind\u00e9x\u00e9s dans <jats:italic>X<\/jats:italic>; on en d\u00e9duit une classification des extensions compl\u00e8tes de <jats:italic>T<jats:sub>x<\/jats:sub><\/jats:italic> en montrant que les applications qui peuvent \u00eatre associ\u00e9es \u00e0 une structure de la forme <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025640_inline2\"\/> pour caract\u00e9riser la th\u00e9orie d'un mod\u00e8le sont d\u00e9termin\u00e9es par leur restriction \u00e0 une partie <jats:italic>M<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025640_inline2\"\/>) d\u00e9finie uniform\u00e9ment, sur laquelle elles prennent des valeurs dans (<jats:bold>N<\/jats:bold> \u2212 {0}) \u22c3 {\u221e}, et que r\u00e9ciproquement toute application de <jats:italic>M<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025640_inline2\"\/>) dans (<jats:bold>N<\/jats:bold> \u2212 {0}) \u22c3 {\u221e} est une telle restriction.<\/jats:p>","DOI":"10.2307\/2274482","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:36:50Z","timestamp":1146940610000},"page":"1192-1212","source":"Crossref","is-referenced-by-count":7,"title":["Th\u00e9ories d'alg\u00e8bres de Boole munies d'id\u00e9aux distingu\u00e9s. II"],"prefix":"10.1017","volume":"55","author":[{"given":"Alain","family":"Touraille","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200025640_ref012","first-page":"1027","volume":"52","author":"Touraille","year":"1987","journal-title":"Th\u00e9ories d'alg\u00e8bres de Boole munies d'id\u00e9aux distingu\u00e9s. I"},{"key":"S0022481200025640_ref011","first-page":"1","article-title":"Decidability of second-order theories and automata on infinite trees","volume":"141","author":"Rabin","year":"1969","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200025640_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BF01980241"},{"key":"S0022481200025640_ref007","unstructured":"Molzan B. , Die Theorie der Booleschen Algebren in der Logik mit Ramsey-Quantor, Dissertation A, Humboldt-Universit\u00e4t, Berlin, 1981."},{"key":"S0022481200025640_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(76)90148-4"},{"key":"S0022481200025640_ref005","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19810272506"},{"key":"S0022481200025640_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0097006"},{"key":"S0022481200025640_ref008","first-page":"186","volume-title":"Proceedings of the third Easter conference on model theory (Gross-K\u00f6ris, 1985)","author":"Molzan","year":"1985"},{"key":"S0022481200025640_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/BF01188389"},{"key":"S0022481200025640_ref003","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-1.1.399"},{"key":"S0022481200025640_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF01163161"},{"key":"S0022481200025640_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8130-3"},{"key":"S0022481200025640_ref010","volume-title":"Memoirs of the American Mathematical Society","author":"Pierce","year":"1972"},{"key":"S0022481200025640_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/BF01188379"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200025640","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T16:24:32Z","timestamp":1558196672000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200025640\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,9]]},"references-count":14,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1990,9]]}},"alternative-id":["S0022481200025640"],"URL":"https:\/\/doi.org\/10.2307\/2274482","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,9]]}}}