{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,8,11]],"date-time":"2022-08-11T23:05:04Z","timestamp":1660259104921},"reference-count":10,"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":5855,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1998,3]]},"abstract":"<jats:p>We prove the undecidability of a very large class of restricted and unrestricted wreath products (Theorem 1.2), and of some skew fields of power series (Section2). Both undecidabilities are obtained by interpreting some enrichments of twisted wreath products, which are themselves proved to be undecidable (Proposition 1.1).<\/jats:p><jats:p>We consider division rings of power series in various languages:<\/jats:p><jats:p>We show (Theorem 2.8) that every power series division ring <jats:italic>k<\/jats:italic>((<jats:italic>B<\/jats:italic>)), whose field of constants <jats:italic>k<\/jats:italic> is commutative and whose ordered group of exponents is noncommutative with a convex center, is undecidable in every extension of the language of rings where the valuation and the ordered group <jats:italic>B<\/jats:italic> are definable.<\/jats:p><jats:p>For certain <jats:italic>k<\/jats:italic> and <jats:italic>B<\/jats:italic> we prove here the undecidability of the 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=\"S0022481200015437_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>where <jats:italic>X<\/jats:italic>\u21be<jats:sub><jats:italic>k((B))xB<\/jats:italic><\/jats:sub> is the restriction of the multiplication to <jats:italic>k<\/jats:italic>((<jats:italic>B<\/jats:italic>)) \u03a7 <jats:italic>B<\/jats:italic>,and \u03b3 is a given conjugation of <jats:italic>k<\/jats:italic>((<jats:italic>B<\/jats:italic>)). This shows that we cannot hope to improve our previous result, a sort of Ax-Kochen-Ershov principle for power series division rings, which ensures that<\/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=\"S0022481200015437_eqnU2\" \/><\/jats:disp-formula><\/jats:p><jats:p>is decidable for every decidable solvable <jats:italic>B<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2586598","type":"journal-article","created":{"date-parts":[[2006,4,18]],"date-time":"2006-04-18T14:43:03Z","timestamp":1145371383000},"page":"237-246","source":"Crossref","is-referenced-by-count":2,"title":["Undecidable wreath products and skew power series fields"],"prefix":"10.1017","volume":"63","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":"S0022481200015437_ref010","unstructured":"Simonetta P. , D\u00e9cidabilit\u00e9 et interpr\u00e9tabilit\u00e9 dans les corps et les groupes non commutatifs, Th\u00e9se , Universit\u00e9 Paris 7, 1994."},{"key":"S0022481200015437_ref003","first-page":"1227","volume":"53","author":"Delon","year":"1988","journal-title":"Ind\u00e9cidabilit\u00e9 de corps de s\u00e9ries formelles"},{"key":"S0022481200015437_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1949-0032593-5"},{"key":"S0022481200015437_ref009","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1951-0041081-0"},{"key":"S0022481200015437_ref001","first-page":"846","article-title":"On the undecidability of power series fields","volume":"16","author":"Ax","year":"1965","journal-title":"Proceedings of the American Mathematical Society"},{"key":"S0022481200015437_ref004","unstructured":"Delon F. and Simonetta P. , Un principe d' Ax-Kochen-Ershov pour des structures int\u00e9rmedials entre groupes et corps valu\u00e9s, to appear in this Journal."},{"key":"S0022481200015437_ref002","volume-title":"Proceedings of the American Mathematical Society","author":"Delon"},{"key":"S0022481200015437_ref008","first-page":"299","volume-title":"Symposium on the theory of models","author":"Robinson","year":"1965"},{"key":"S0022481200015437_ref006","doi-asserted-by":"publisher","DOI":"10.2307\/2372087"},{"key":"S0022481200015437_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF01234910"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200015437","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,11]],"date-time":"2019-05-11T15:03:41Z","timestamp":1557587021000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200015437\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,3]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1998,3]]}},"alternative-id":["S0022481200015437"],"URL":"https:\/\/doi.org\/10.2307\/2586598","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,3]]}}}