{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T10:31:49Z","timestamp":1772447509342,"version":"3.50.1"},"reference-count":7,"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":8685,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,6]]},"abstract":"<jats:p>In what follows, a <jats:italic>coset<\/jats:italic> is a subset of a group <jats:italic>G<\/jats:italic> of the form <jats:italic>aH<\/jats:italic>, where <jats:italic>H<\/jats:italic> is a subgroup of <jats:italic>G<\/jats:italic>; <jats:italic>H<\/jats:italic> can be recovered from the coset <jats:italic>C<\/jats:italic>: it is the only subgroup which is obtained from <jats:italic>C<\/jats:italic> by a left translation; we note in passing that these cosets, that we write systematically with the group to the right, are also of the form <jats:italic>Ka<\/jats:italic>, since <jats:italic>aH<\/jats:italic> = <jats:italic>aHa<jats:sup>\u22121<\/jats:sup>a<\/jats:italic>. A classical combinatorial lemma involving cosets appears in Neumann [1952]: <jats:italic>If the coset C<\/jats:italic> = <jats:italic>aH is the union of the finite family of cosets C<\/jats:italic><jats:sub>1<\/jats:sub> = <jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub><jats:italic>H<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026,<jats:italic>C<jats:sub>n<\/jats:sub> = a<jats:sub>n<\/jats:sub>H<jats:sub>n<\/jats:sub>, then it is the union of those C<jats:sub>i<\/jats:sub> whose corresponding H<jats:sub>i<\/jats:sub> has finite index in H<\/jats:italic>.<\/jats:p><jats:p>In a structure where a group <jats:italic>G<\/jats:italic> is defined, Boolean combinations of cosets modulo its definable subgroups form a family of definable sets (by definable, we mean \u201cdefinable with parameters\u201d). The situation when any definable set is of that kind has been characterized model-theoretically in Hrushovski and Pillay [1987]: <jats:italic>A group G is one-based if and only if, for each n, every definable subset of the cartesian power G<jats:sup>n<\/jats:sup> is a(finite!) Boolean combination of cosets modulo definable subgroups<\/jats:italic>. One side is given by a beautiful lemma of Pillay, stating that, in a one-based group which is saturated enough, every type is a right translate of the generic of its left stabilizer.<\/jats:p>","DOI":"10.2307\/2274656","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:35:29Z","timestamp":1146954929000},"page":"670-673","source":"Crossref","is-referenced-by-count":5,"title":["The last word on elimination of quantifiers in modules"],"prefix":"10.1017","volume":"55","author":[{"given":"Hans B.","family":"Gute","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"K. K.","family":"Reuter","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200026062_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(84)90014-9"},{"key":"S0022481200026062_ref006","doi-asserted-by":"publisher","DOI":"10.4064\/fm-41-2-203-271"},{"key":"S0022481200026062_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(09)70557-9"},{"key":"S0022481200026062_ref003","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-27.2.247"},{"key":"S0022481200026062_ref004","volume-title":"Cours de th\u00e9orie des mod\u00e8les","author":"Poizat","year":"1985"},{"key":"S0022481200026062_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(09)70556-7"},{"key":"S0022481200026062_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02756561"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200026062","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T21:13:22Z","timestamp":1558214002000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200026062\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,6]]},"references-count":7,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1990,6]]}},"alternative-id":["S0022481200026062"],"URL":"https:\/\/doi.org\/10.2307\/2274656","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,6]]}}}