{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,20]],"date-time":"2025-10-20T10:07:37Z","timestamp":1760954857148},"reference-count":13,"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":5306,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1999,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>An Ax-Kochen-Ershov principle for intermediate structures between valued groups and valued fields.<\/jats:p><jats:p>We will consider structures that we call valued <jats:italic>B<\/jats:italic>-groups and which are of the form \u3008<jats:italic>G, B<\/jats:italic>, *, \u03c5\u3009 where<\/jats:p><jats:p>\u2013 <jats:italic>G<\/jats:italic> is an abelian group,<\/jats:p><jats:p>\u2013 <jats:italic>B<\/jats:italic> is an ordered group,<\/jats:p><jats:p>\u2013 \u03c5 is a valuation denned on <jats:italic>G<\/jats:italic> taking its values in <jats:italic>B<\/jats:italic>,<\/jats:p><jats:p>\u2013 * is an action of <jats:italic>B<\/jats:italic> on <jats:italic>G<\/jats:italic> satisfying: \u2200<jats:italic>x<\/jats:italic> \u03f5 <jats:italic>G<\/jats:italic> \u2200 <jats:italic>b<\/jats:italic> \u2208 <jats:italic>B<\/jats:italic> \u03c5(<jats:italic>x<\/jats:italic> * <jats:italic>b<\/jats:italic>) = \u03bd(<jats:italic>x<\/jats:italic>) \u00b7 <jats:italic>b<\/jats:italic>.<\/jats:p><jats:p>The analysis of Kaplanski for valued fields can be adapted to our context and allows us to formulate an Ax-Kochen-Ershov principle for valued <jats:italic>B<\/jats:italic>-groups: we axiomatise those which are in some sense existentially closed and also obtain many of their model-theoretical properties. Let us mention some applications:<\/jats:p><jats:p>1. Assume that \u03c5(<jats:italic>x<\/jats:italic>) = \u03c5(<jats:italic>nx<\/jats:italic>) for every integer <jats:italic>n<\/jats:italic> \u2260 0 and <jats:italic>x<\/jats:italic> \u03f5 <jats:italic>G, B<\/jats:italic> is solvable and acts on <jats:italic>G<\/jats:italic> in such a way that, for the induced action, Z[<jats:italic>B<\/jats:italic>] \u2216 {0} embeds in the automorphism group of <jats:italic>G<\/jats:italic>. Then \u3008<jats:italic>G, B<\/jats:italic>, *, \u03c5\u3009 is decidable if and only if <jats:italic>B<\/jats:italic> is decidable as an ordered group.<\/jats:p><jats:p><jats:italic>2.<\/jats:italic> Given a field <jats:italic>k<\/jats:italic> and an ordered group <jats:italic>B<\/jats:italic>, we consider the generalised power series field <jats:italic>k<\/jats:italic>((<jats:italic>B<\/jats:italic>)) endowed with its canonical valuation. We consider also the following structure:<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200013177_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>where <jats:italic>k<\/jats:italic>((<jats:italic>B<\/jats:italic>))<jats:sub>+<\/jats:sub> is the additive group of <jats:italic>k<\/jats:italic>((<jats:italic>B<\/jats:italic>)), <jats:italic>S<\/jats:italic> is a unary predicate interpreting {<jats:italic>T<jats:sup>b<\/jats:sup> \u2223 b<\/jats:italic> \u03f5<jats:italic>B<\/jats:italic>}, and \u00d7\u21be<jats:sub><jats:italic>k<\/jats:italic>((<jats:italic>B<\/jats:italic>))\u00d7<jats:italic>S<\/jats:italic><\/jats:sub> is the multiplication restricted to <jats:italic>k<\/jats:italic>((<jats:italic>B<\/jats:italic>)) \u00d7 <jats:italic>S<\/jats:italic>, structure which is a reduct of the valued field <jats:italic>k<\/jats:italic>((<jats:italic>B<\/jats:italic>)) with its canonical cross section. Then our result implies that if <jats:italic>B<\/jats:italic> is solvable and decidable as an ordered group, then <jats:bold>M<\/jats:bold> is decidable.<\/jats:p><jats:p>3. A valued <jats:italic>B<\/jats:italic>\u2013group has a residual group and our Ax-Kochen-Ershov principle remains valid in the context of expansions of residual group and value group. In particular, by adding a residual order we obtain new examples of solvable ordered groups having a decidable theory.<\/jats:p>","DOI":"10.2307\/2586616","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T14:03:11Z","timestamp":1146924191000},"page":"991-1027","source":"Crossref","is-referenced-by-count":1,"title":["Un principe d'ax-kochen-ershov pour des structures interm\u00e9diates entre groupes et corps valu\u00e9s"],"prefix":"10.1017","volume":"64","author":[{"given":"Fran\u00e7oise","family":"Delon","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Patrick","family":"Simonetta","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200013177_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-97-03912-9"},{"key":"S0022481200013177_ref007","volume-title":"Th\u00e9orie des valuations","author":"Ribenboim","year":"1964"},{"key":"S0022481200013177_ref006","volume-title":"The algebraic structure of group rings","author":"Passman","year":"1985"},{"key":"S0022481200013177_ref012","first-page":"60","volume":"62","author":"Simonetta","year":"1997","journal-title":"Une correspondance entre anneaux partiels et groupes"},{"key":"S0022481200013177_ref008","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/004"},{"key":"S0022481200013177_ref009","first-page":"173","article-title":"A theorem on quantifier elimination","volume":"5","author":"Shoenfield","year":"1971","journal-title":"Symposia Mathematica"},{"key":"S0022481200013177_ref003","first-page":"237","volume":"63","author":"Delon","year":"1998","journal-title":"Undecidable wreath products and skew power series fields"},{"key":"S0022481200013177_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-87081-1"},{"key":"S0022481200013177_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1949-0032593-5"},{"key":"S0022481200013177_ref010","unstructured":"Simonetta P. , Equivalence \u00e9l\u00e9mentaire et d\u00e9cidabilit\u00e9 pour des structures du type groupe agissant sur un groupe ab\u00e9lien, to appear in this Journal."},{"key":"S0022481200013177_ref011","unstructured":"Simonetta P. , D\u00e9cidabilit\u00e9 et interpr\u00e9tabilit\u00e9 dans les corps et les groupes non commutatifs, Th\u00e8se de l'Universit\u00e9 Paris 7 , 1994."},{"key":"S0022481200013177_ref004","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-42-00922-0"},{"key":"S0022481200013177_ref001","volume-title":"Alg\u00e8bre commutative","author":"Bourbaki","year":"1964"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200013177","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,9]],"date-time":"2019-05-09T16:38:02Z","timestamp":1557419882000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200013177\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,9]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1999,9]]}},"alternative-id":["S0022481200013177"],"URL":"https:\/\/doi.org\/10.2307\/2586616","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1999,9]]}}}