{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,19]],"date-time":"2023-10-19T05:15:53Z","timestamp":1697692553033},"reference-count":37,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2003,1,21]],"date-time":"2003-01-21T00:00:00Z","timestamp":1043107200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2003,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The new result of this paper is that for <jats:italic>\u03b8<\/jats:italic>(<jats:bold><jats:italic>x<\/jats:italic><\/jats:bold>; <jats:bold><jats:italic>a<\/jats:italic><\/jats:bold>)\u2010stable (a weakening of \u201c<jats:italic>T<\/jats:italic> is stable\u201d) we have <jats:italic>S<\/jats:italic>1[<jats:italic>\u03b8<\/jats:italic>(<jats:bold><jats:italic>x<\/jats:italic><\/jats:bold>; <jats:bold><jats:italic>a<\/jats:italic><\/jats:bold>)] = <jats:italic>D<\/jats:italic>[<jats:italic>\u03b8<\/jats:italic>(<jats:bold><jats:italic>x<\/jats:italic><\/jats:bold>; <jats:bold><jats:italic>a<\/jats:italic><\/jats:bold>), <jats:italic>L<\/jats:italic>, \u221e]. <jats:italic>S<\/jats:italic>1 is Hrushovski's rank. This is an improvement of a result of Kim and Pillay, who for simple theories under the (strong) assumption that either of the ranks be finite obtained the same identity. Only the first equality is new, the second equality is a result of Shelah from the seventies. We derive it by studying localizations of several rank functions, we get the following<\/jats:p><jats:p> Main Theorem. Suppose that <jats:italic>\u03bc<\/jats:italic> is regular satisfying <jats:italic>\u03bc<\/jats:italic> \u2265 |<jats:italic>T<\/jats:italic>|<jats:sup>+<\/jats:sup>, <jats:italic>p<\/jats:italic> is a finite type, and \u0394 is a set of formulas closed under Boolean operations. If either (a) <jats:italic>R<\/jats:italic>[<jats:italic>p<\/jats:italic>, \u0394, <jats:italic>\u03bc<\/jats:italic><jats:sup>+<\/jats:sup>] &lt; \u221e or (b) <jats:italic>p<\/jats:italic> is \u0394\u2010stable and <jats:italic>\u03bc<\/jats:italic> satisfies \u201cfor every sequence {<jats:italic>\u03bc<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> : <jats:italic>i<\/jats:italic> &lt; |\u0394| + \u2135<jats:sub>0<\/jats:sub>} of cardinals <jats:italic>\u03bc<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> &lt; <jats:italic>\u03bc<\/jats:italic> we have that <jats:styled-content>$ \\prod _{i &lt; \\vert \\Delta \\vert + \\aleph _{0}} \\mu _{i} &lt; \\mu $<\/jats:styled-content> holds\u201d, then <jats:italic>S<\/jats:italic>[<jats:italic>p<\/jats:italic>, \u0394, <jats:italic>\u03bc<\/jats:italic><jats:sup>+<\/jats:sup>] = <jats:italic>D<\/jats:italic>[<jats:italic>p<\/jats:italic>, \u0394, <jats:italic>\u03bc<\/jats:italic><jats:sup>+<\/jats:sup>] = <jats:italic>R<\/jats:italic>[<jats:italic>p<\/jats:italic>, \u0394, <jats:italic>\u03bc<\/jats:italic><jats:sup>+<\/jats:sup>].<\/jats:p><jats:p>The <jats:italic>S<\/jats:italic> rank above is a localized version of Hrushovski's <jats:italic>S<\/jats:italic>1 rank. This rank, as well as our systematic use of local stability, allows us to get a more conceptual proof of the equality of <jats:italic>D<\/jats:italic> and <jats:italic>R<\/jats:italic>, which is an old result of Shelah. A particular (asymptotic) case of the theorem offers a new sufficient condition for the equality of <jats:italic>S<\/jats:italic>1 and <jats:italic>D<\/jats:italic>[\u00b7, <jats:italic>L<\/jats:italic>, \u221e]. We also manage, due to a more general approach, to avoid some combinatorial difficulties present in Shelah's original exposition.<\/jats:p>","DOI":"10.1002\/malq.200310012","type":"journal-article","created":{"date-parts":[[2003,1,24]],"date-time":"2003-01-24T16:13:50Z","timestamp":1043424830000},"page":"115-128","source":"Crossref","is-referenced-by-count":0,"title":["The equality <i>S<\/i>1 = <i>D<\/i> = <i>R<\/i>"],"prefix":"10.1002","volume":"49","author":[{"given":"Rami","family":"Grossberg","sequence":"first","affiliation":[]},{"given":"Alexei","family":"Kolesnikov","sequence":"additional","affiliation":[]},{"given":"Ivan","family":"Toma\u0161i\u0107","sequence":"additional","affiliation":[]},{"given":"Monica","family":"Van Dieren","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2003,1,21]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.2307\/1996618"},{"key":"e_1_2_1_3_2","doi-asserted-by":"crossref","unstructured":"J. T.Baldwin Fundamentals of Stability Theory (Springer\u2010Verlag Berlin et al. 1988).","DOI":"10.1007\/978-3-662-07330-8"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(74)90018-7"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.2307\/2271517"},{"key":"e_1_2_1_6_2","doi-asserted-by":"crossref","unstructured":"E.Bouscaren An introduction to independence and local modularity. In: Algebraic Model Theory (B. Hart A. Lachlan and M. Valeriote eds.) NATO ASI Series Mathematical and Physical Sciences Vol. 496 (Kluwer Academic Publishers 1997).","DOI":"10.1007\/978-94-015-8923-9_1"},{"key":"e_1_2_1_7_2","doi-asserted-by":"crossref","unstructured":"S.Buechler Essential Stability Theory (Springer\u2010Verlag Berlin et al. 1996).","DOI":"10.1007\/978-3-642-80177-8"},{"key":"e_1_2_1_8_2","unstructured":"C. C.ChangandH. J.Keisler Model Theory (North\u2010Holland Publishing Co. Amsterdam 1990)."},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-99-02498-8"},{"key":"e_1_2_1_10_2","unstructured":"R.Grossberg A Course in Model Theory. Book in preparation."},{"key":"e_1_2_1_11_2","unstructured":"R.Grossbberg J.Iovino andO.Lessmann A primer of simple theories. Archive Math. Logic (to appear)."},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1007\/s001530050157"},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(84)90005-8"},{"key":"e_1_2_1_14_2","unstructured":"E.Hrushovski Pseudofinite fields and related structures. Unpublished Manuscript."},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02758643"},{"key":"e_1_2_1_16_2","first-page":"69","article-title":"Definable groups of algebraic groups over finite fields","volume":"462","author":"Hrushovski E.","year":"1995","journal-title":"Crelle's Journal"},{"key":"e_1_2_1_17_2","unstructured":"E.HrushovskiandTh.Scanlon Lascar and Morley ranks differ in differentially closed fields. Preprint."},{"key":"e_1_2_1_18_2","unstructured":"Th.Jech Set Theory (Academic Press New York 1978)."},{"issue":"2","key":"e_1_2_1_19_2","first-page":"57","article-title":"Forking in simple unstable theories","author":"Kim B.","year":"1998","journal-title":"J. London Math. Soc."},{"key":"e_1_2_1_20_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(97)00019-5"},{"key":"e_1_2_1_21_2","first-page":"A1233","article-title":"Types d\u00e9finissables et produit de types","volume":"272","author":"Lascar D.","year":"1971","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. A\u2010B"},{"key":"e_1_2_1_22_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757234"},{"key":"e_1_2_1_23_2","unstructured":"D.Lascar Stability in Model Theory (John Wiley & Sons Inc. New York 1987)."},{"key":"e_1_2_1_24_2","doi-asserted-by":"publisher","DOI":"10.2307\/2273127"},{"key":"e_1_2_1_25_2","doi-asserted-by":"crossref","first-page":"58","DOI":"10.4064\/cm-3-1-58-62","article-title":"On the categoricity in power of elementary deductive systems and related problems","volume":"3","author":"\u0141o\u015b J.","year":"1954","journal-title":"Colloq. Math."},{"key":"e_1_2_1_26_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760649"},{"key":"e_1_2_1_27_2","unstructured":"W.Marsh On\u03c91\u2010categorical and not\u03c9\u2010categorical theories. PhD Thesis Dartmouth College 1966."},{"key":"e_1_2_1_28_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1965-0175782-0"},{"key":"e_1_2_1_29_2","unstructured":"A.Pillay An Introduction to Stability Theory (Clarendon Press Oxford University Press New York 1983)."},{"key":"e_1_2_1_30_2","unstructured":"B.Poizat Suites d'indiscernables dans les th\u00e9ories stables. Technical Report Univ. Pierre et Marie Curie Paris 1978."},{"key":"e_1_2_1_31_2","unstructured":"G.Sacks Saturated Model Theory (W. A. Benjamin Inc. Reading 1972)."},{"key":"e_1_2_1_32_2","doi-asserted-by":"publisher","DOI":"10.2307\/2274286"},{"key":"e_1_2_1_33_2","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90007-0"},{"key":"e_1_2_1_34_2","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90015-5"},{"key":"e_1_2_1_35_2","doi-asserted-by":"crossref","unstructured":"S.Shelah Categoricity of uncountable theories. In: Proceedings of the Tarski Symposium (L. A. Henkin et al. eds.) Providence (R. I.) 1974 pp. 187 \u2013 203.","DOI":"10.1090\/pspum\/025\/0373874"},{"key":"e_1_2_1_36_2","unstructured":"S.Shelah Classification Theory and the Number of Nonisomorphic Models (North\u2010Holland Publishing Co. Amsterdam 1978)."},{"key":"e_1_2_1_37_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02761994"},{"key":"e_1_2_1_38_2","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(80)90009-1"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.200310012","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.200310012","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,18]],"date-time":"2023-10-18T13:40:41Z","timestamp":1697636441000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.200310012"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1,21]]},"references-count":37,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2003,3]]}},"alternative-id":["10.1002\/malq.200310012"],"URL":"https:\/\/doi.org\/10.1002\/malq.200310012","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,1,21]]}}}