{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T11:08:36Z","timestamp":1777460916009,"version":"3.51.4"},"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":1380,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2010,6]]},"abstract":"<jats:p>Dans [2], Salma et Franz Viktor Kuhlmann d\u00e9finissent plu-sieurs notions d'espace vectoriel valu\u00e9 et d\u00e9montrent un principe d'Ax-Kochen pour celles de ces structures dans lesquelles la multiplication par un scalaire du corps pr\u00e9serve la valuation. Nous travaillons ici avec des conditions plus faibles. On va d\u00e9finir en premier lieu la notion d'espace vectoriel valu\u00e9 sur un corps <jats:italic>K<\/jats:italic>, et associer canoniquement \u00e0 un tel espace une valuation sur <jats:italic>K<\/jats:italic>, not\u00e9e <jats:italic>w<\/jats:italic>, compatible en un sens \u00e0 pr\u00e9ciser avec celle de l'espace vectoriel. Ceci va nous permettre de parler de (<jats:italic>K, w<\/jats:italic>)-espaces vectoriels valu\u00e9s pour tout corps valu\u00e9 (<jats:italic>K, w<\/jats:italic>) (definition 24). Le cas o\u00f9 (<jats:italic>K, w<\/jats:italic>) est trivialement valu\u00e9 correspond exactement \u00e0 la condition de [2] de pr\u00e9servation de la valuation par la multiplication par un scalaire. On fixe un corps <jats:italic>K<\/jats:italic>. Les espaces vectoriels valu\u00e9s sur le corps <jats:italic>K<\/jats:italic> vont \u00eatre trait\u00e9s comme des structures \u00e0 deux sortes: la sorte de l'espace vectoriel et celle de l'espace des valuations. Le langage <jats:italic>L<jats:sub>E<\/jats:sub><\/jats:italic> de la sorte de l'espace vectoriel contient le symbole de la somme, celui de l'\u00e9l\u00e9ment neutre, et un symbole de fonction unaire pour chaque scalaire de <jats:italic>K<\/jats:italic>. Le langage <jats:italic>L<jats:sub>V<\/jats:sub><\/jats:italic> de l'espace des valuations contient une relation d'ordre total, un symbole de fonction unaire pour la multiplication par chaque scalaire de <jats:italic>K<\/jats:italic>, et pour tout <jats:italic>n<\/jats:italic> \u2208 \u2115*, un symbole de pr\u00e9dicat <jats:italic>R<jats:sub>n<\/jats:sub><\/jats:italic> contr\u00f4lant la cardinalit\u00e9 r\u00e9siduelle. <jats:italic>L<jats:sub>V<\/jats:sub><\/jats:italic> contient aussi un symbole de fonction unaire <jats:italic>s<\/jats:italic> qui sera la fonction successeur quand elle est d\u00e9finie et l'identit\u00e9 sinon.<\/jats:p>","DOI":"10.2178\/jsl\/1268917502","type":"journal-article","created":{"date-parts":[[2010,3,18]],"date-time":"2010-03-18T09:05:57Z","timestamp":1268903157000},"page":"741-758","source":"Crossref","is-referenced-by-count":1,"title":["Espaces vectoriels <i>C<\/i>-minimaux"],"prefix":"10.1017","volume":"75","author":[{"given":"Fares","family":"Maalouf","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200002759_ref006","first-page":"19","article-title":"Quantifier elimination for linear formulas over ordered and valued fields","volume":"33","author":"van den Dries","year":"1981","journal-title":"Bulletin de la Soci\u00e9t\u00e9 Math\u00e9matique de Belgique. S\u00e9rie B"},{"key":"S0022481200002759_ref005","volume-title":"Model theory: An introduction","author":"Marker","year":"2002"},{"key":"S0022481200002759_ref003","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/0168-0072(94)90064-7","article-title":"Cell decompositions of C-minimal structures","volume":"66","author":"Macpherson","year":"1994","journal-title":"Annals of Pure and Applied Logic"},{"key":"S0022481200002759_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(95)00037-2"},{"key":"S0022481200002759_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-5640-0_10"},{"key":"S0022481200002759_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/memo\/0623"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200002759","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,28]],"date-time":"2019-04-28T15:37:47Z","timestamp":1556465867000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200002759\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,6]]},"references-count":6,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2010,6]]}},"alternative-id":["S0022481200002759"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1268917502","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,6]]}}}