{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,21]],"date-time":"2024-07-21T02:48:56Z","timestamp":1721530136754},"reference-count":3,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":4119,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,12]]},"abstract":"<jats:p>Borovik proposed an axiomatic treatment of Morley rank in groups, later modified by Poizat, who showed that in the context of groups the resulting notion of rank provides a characterization of groups of finite Morley rank [2]. (This result makes use of ideas of Lascar, which it encapsulates in a neat way.) These axioms form the basis of the algebraic treatment of groups of finite Morley rank undertaken in [1].<\/jats:p><jats:p>There are, however, ranked structures, i.e., structures on which a Borovik-Poizat rank function is defined, which are not \u2135<jats:sub>0<\/jats:sub>-stable [1, p. 376]. In [2, p. 9] Poizat raised the issue of the relationship between this notion of rank and stability theory in the following terms: \u201c\u2026 un<jats:italic>groupe<\/jats:italic>de Borovik est une structure stable, alors qu'un univers rang\u00e9 n'a aucune raison de l'\u00eatre \u2026\u201d (emphasis added). Nonetheless, we will prove the following:<\/jats:p><jats:p>Theorem 1.1.<jats:italic>A ranked structure is superstable<\/jats:italic>.<\/jats:p><jats:p>An example of a non-\u2135<jats:sub>0<\/jats:sub>-stable structure with Borovik-Poizat rank 2 is given in [1, p. 376]. Furthermore, it appears that this example can be modified in a straightforward way to give \u2135<jats:sub>0<\/jats:sub>-stable structures of Borovik-Poizat rank 2 in which the Morley rank is any countable ordinal (which would refute a claim of [1, p. 373, proof of C.4]). We have not checked the details. This does not leave much room for strenghthenings of our theorem. On the other hand, the proof of Theorem 1.1 does give a finite bound for the heights of certain trees of definable sets related to unsuperstability, as we will see in Section 5.<\/jats:p>","DOI":"10.2178\/jsl\/1190150299","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T19:16:54Z","timestamp":1197573414000},"page":"1570-1578","source":"Crossref","is-referenced-by-count":2,"title":["Borovik-Poizat rank and stability"],"prefix":"10.1017","volume":"67","author":[{"given":"Jeffrey","family":"Burdges","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gregory","family":"Cherlin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120000921X_ref002","volume-title":"Groupes stables","author":"Poizat","year":"1987"},{"key":"S002248120000921X_ref003","volume-title":"Classification Theory and the Number of Nonisomorphic Models","author":"Shelah","year":"1990"},{"key":"S002248120000921X_ref001","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198534457.001.0001","volume-title":"Groups of Finite Morley Rank","author":"Borovik","year":"1994"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120000921X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,20]],"date-time":"2024-02-20T10:42:38Z","timestamp":1708425758000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120000921X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,12]]},"references-count":3,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2002,12]]}},"alternative-id":["S002248120000921X"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1190150299","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,12]]}}}