{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T16:25:43Z","timestamp":1767198343772,"version":"build-2238731810"},"reference-count":7,"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":8228,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1991,9]]},"abstract":"\n Here we consider some problems concerning regular types. In the first place we consider a strongly minimal set\n D<\/jats:italic>\n . One can ask what is the strength of the assumption that\n D<\/jats:italic>\n has (full) elimination of imaginaries (namely, every definable set\n X<\/jats:italic>\n over\n D<\/jats:italic>\n has as canonical parameter some tuple from\n D<\/jats:italic>\n ). We show that\n D<\/jats:italic>\n cannot be locally modular. Nontriviality of\n D<\/jats:italic>\n is immediate. However, to exclude the locally modular nontrivial case one has to understand structures of the form\n G\/E<\/jats:italic>\n , where\n G<\/jats:italic>\n is a modular strongly minimal group and\n E<\/jats:italic>\n is a definable equivalence relation on\n G<\/jats:italic>\n with finite classes. We show that the quotient structure\n G\/E<\/jats:italic>\n can be obtained in two steps. First quotient by a finite subgroup\n K<\/jats:italic>\n of\n G<\/jats:italic>\n to obtain a strongly minimal group\n H<\/jats:italic>\n . Now let \u0393 be a finite subgroup of the group Aff(\n H<\/jats:italic>\n ) of definable affine automorphisms of\n H<\/jats:italic>\n (namely maps of the form\n x<\/jats:italic>\n \u2192 \u03b1\n x<\/jats:italic>\n +\n a<\/jats:italic>\n , where \u03b1 is a definable automorphism of\n H<\/jats:italic>\n and\n a<\/jats:italic>\n \u2208\n H<\/jats:italic>\n ), and quotient\n H<\/jats:italic>\n by \u0393 (namely form the orbit space of\n H<\/jats:italic>\n under \u0393). It can clearly be arranged that \u0393 contains no nontrivial subgroup of translations.\n <\/jats:p>\n \n In the second place we look at a nontrivial modular regular type\n p<\/jats:italic>\n whose pregeometry is actually a geometry. The geometry is then known to be (infinite-dimensional) projective geometry over a division ring\n F<\/jats:italic>\n . We ask whether\n F<\/jats:italic>\n is definable (internally to\n p<\/jats:italic>\n ). If\n F<\/jats:italic>\n is finite, this is clear. In fact in this case\n p<\/jats:italic>\n must have\n U<\/jats:italic>\n -rank 1. So we assume\n F<\/jats:italic>\n to be infinite. We are only able to show definability of\n F<\/jats:italic>\n in the case where\n F<\/jats:italic>\n is a field, using some results on 2-transitive subgroups of PGL [V]. Moreover in the superstable case we also observe that\n p<\/jats:italic>\n is isolated.\n <\/jats:p>","DOI":"10.2307\/2275067","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:41:03Z","timestamp":1146940863000},"page":"1003-1011","source":"Crossref","is-referenced-by-count":2,"title":["Some remarks on modular regular types"],"prefix":"10.1017","volume":"56","author":[{"given":"Anand","family":"Pillay","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200024154_ref004","first-page":"709","volume":"51","author":"Pillay","year":"1986","journal-title":"Some remarks on definable equivalence relations in O-minimal structures"},{"key":"S0022481200024154_ref003","first-page":"250","volume":"56","author":"Loveys","year":"1991","journal-title":"Abelian groups with modular generic"},{"key":"S0022481200024154_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0082236"},{"key":"S0022481200024154_ref005","first-page":"858","volume":"54","author":"Pillay","year":"1989","journal-title":"A note on subgroups of the automorphism group of a saturated model, and regular types"},{"key":"S0022481200024154_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(09)70556-7"},{"key":"S0022481200024154_ref006","first-page":"1151","volume":"48","author":"Poizat","year":"1983","journal-title":"Une th\u00e9orie de Galois imaginaire"},{"key":"S0022481200024154_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(82)90084-9"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200024154","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T06:35:00Z","timestamp":1679466900000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200024154\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1991,9]]},"references-count":7,"aliases":["10.2178\/jsl\/1183743746"],"journal-issue":{"issue":"3","published-print":{"date-parts":[[1991,9]]}},"alternative-id":["S0022481200024154"],"URL":"https:\/\/doi.org\/10.2307\/2275067","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1991,9]]}}}