{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,2,4]],"date-time":"2024-02-04T07:40:19Z","timestamp":1707032419290},"reference-count":5,"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":5124,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2000,3]]},"abstract":"<jats:p><jats:italic>CM<\/jats:italic>-triviality of a stable theory is a notion introduced by Hrushovski [1]. The importance of this property is first that it holds of Hrushovski's new non 1-based strongly minimal sets, and second that it is still quite a restrictive property, and forbids the existence of definable fields or simple groups (see [2]). In [5], Frank Wagner posed some questions about<jats:italic>CM<\/jats:italic>-triviality, asking in particular whether a structure of finite rank, which is \u201ccoordinatized\u201d by<jats:italic>CM<\/jats:italic>-trivial types of rank 1, is itself<jats:italic>CM<\/jats:italic>-trivial. (Actually Wagner worked in a slightly more general context, adapting the definitions to a certain \u201clocal\u201d framework, in which algebraic closure is replaced by<jats:italic>P<\/jats:italic>-closure, for<jats:italic>P<\/jats:italic>some family of types. We will, however, remain in the standard context, and will just remark here that it is routine to translate our results into Wagner's framework, as well as to generalise to the superstable theory\/regular type context.) In any case we answer Wagner's question positively. Also in an attempt to put forward some concrete conjectures about the possible geometries of strongly minimal sets (or stable theories) we tentatively suggest a hierarchy of geometric properties of forking, the first two levels of which correspond to 1-basedness and<jats:italic>CM<\/jats:italic>-triviality respectively. We do not know whether this is a strict hierarchy (or even whether these are the \u201cright\u201d notions), but we conjecture that it is, and moreover that a counterexample to Cherlin's conjecture can be found at level three in the hierarchy.<\/jats:p>","DOI":"10.2307\/2586549","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:01:31Z","timestamp":1146938491000},"page":"474-480","source":"Crossref","is-referenced-by-count":7,"title":["A note on<i>CM<\/i>-triviality and the geometry of forking"],"prefix":"10.1017","volume":"65","author":[{"given":"Anand","family":"Pillay","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200012597_ref005","author":"Wagner","journal-title":"CM-triviality and stable groups"},{"key":"S0022481200012597_ref004","volume-title":"Groupes stables","author":"Poizat","year":"1987"},{"key":"S0022481200012597_ref003","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198534372.001.0001","volume-title":"Geometric Stability Theory","author":"Pillay","year":"1996"},{"key":"S0022481200012597_ref002","first-page":"1251","volume":"60","author":"Pillay","year":"1995","journal-title":"The geometry of forking and groups of finite Morley rank"},{"key":"S0022481200012597_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(93)90171-9"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200012597","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,4]],"date-time":"2024-02-04T07:19:15Z","timestamp":1707031155000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200012597\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,3]]},"references-count":5,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2000,3]]}},"alternative-id":["S0022481200012597"],"URL":"https:\/\/doi.org\/10.2307\/2586549","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,3]]}}}