{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,2,4]],"date-time":"2024-02-04T07:40:28Z","timestamp":1707032428198},"reference-count":8,"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>In this paper we construct a non-<jats:italic>CM<\/jats:italic>-trivial stable theory in which no infinite field is interpretable. In fact our theory will also be trivial and \u03c9-stable, but of infinite Morley rank. A long term aim would be to find a non<jats:italic>CM<\/jats:italic>-trivial theory which has finite Morley rank (or is even strongly minimal) and does not interpret a field. The construction in this paper is direct, and is a \u201c3-dimensional\u201d version of the free pseudoplane. In a sense we are cheating: the original point of the notion of<jats:italic>CM<\/jats:italic>-triviality was to describe the geometry of a strongly minimal set, or even of a regular type. In our example, non-<jats:italic>CM<\/jats:italic>-triviality will come from the behaviour of three orthogonal regular types.<\/jats:p><jats:p>A stable theory is said to be<jats:italic>CM<\/jats:italic>-trivial if whenever<jats:italic>A<\/jats:italic>\u2286<jats:italic>B<\/jats:italic>and acl(<jats:italic>Ac<\/jats:italic>) \u2229 acl(<jats:italic>B<\/jats:italic>) = acl(<jats:italic>A<\/jats:italic>) in<jats:italic>T<\/jats:italic><jats:sup>eq<\/jats:sup>, then Cb(stp(<jats:italic>c<\/jats:italic>\/<jats:italic>A<\/jats:italic>)) \u2286 Cb(stp(<jats:italic>c<\/jats:italic>\/<jats:italic>B<\/jats:italic>)). ( An infinite stable field will not be<jats:italic>CM<\/jats:italic>-trivial.) The notion is due to Hrushovski [3], where he gave several equivalent definitions, as well as showing that his new strongly minimal sets constructed \u201cab ovo\u201d were<jats:italic>CM<\/jats:italic>-trivial. The notion was studied further in [6] where it was shown that<jats:italic>CM<\/jats:italic>-trivial groups of finite Morley rank are nilpotent-by-finite. These results were generalized in various ways to the superstable case in [8].<\/jats:p>","DOI":"10.2307\/2586547","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:01:31Z","timestamp":1146938491000},"page":"443-460","source":"Crossref","is-referenced-by-count":6,"title":["A free pseudospace"],"prefix":"10.1017","volume":"65","author":[{"given":"Andreas","family":"Baudisch","sequence":"first","affiliation":[]},{"given":"Anand","family":"Pillay","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200012573_ref008","author":"Wagner","journal-title":"CM-triviality and stable groups"},{"key":"S0022481200012573_ref006","first-page":"1251","volume":"60","author":"Pillay","year":"1995","journal-title":"The geometry of forking and groups of finite Morley rank"},{"key":"S0022481200012573_ref005","author":"Pillay","journal-title":"CM-triviality and the geometry of forking"},{"key":"S0022481200012573_ref004","unstructured":"Hrushovski E. and Srour G. , On stable non-equational theories, manuscript, 1989."},{"key":"S0022481200012573_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(93)90171-9"},{"key":"S0022481200012573_ref002","unstructured":"Baudisch A. , Mekler's construction preserves CM-triviality, preprint, 1997."},{"key":"S0022481200012573_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-96-01623-6"},{"key":"S0022481200012573_ref007","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198534372.001.0001","volume-title":"Geometric stability theory","author":"Pillay","year":"1996"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200012573","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\/S0022481200012573\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,3]]},"references-count":8,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2000,3]]}},"alternative-id":["S0022481200012573"],"URL":"https:\/\/doi.org\/10.2307\/2586547","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,3]]}}}