{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T22:59:35Z","timestamp":1771455575872,"version":"3.50.1"},"reference-count":24,"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":923,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2011,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We continue our study of finitary abstract elementary classes, defined in [7]. In this paper, we prove a categoricity transfer theorem for a case of simple finitary AECs. We introduce the concepts of weak<jats:italic>\u03ba<\/jats:italic>-categoricity and f-primary models to the framework of \u2135<jats:sub>0<\/jats:sub>-stable simple finitary AECs with the extension property, whereby we gain the following theorem: Let (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200001523_inline1\"\/>) be a simple finitary AEC, weakly categorical in some uncountable<jats:italic>\u03ba<\/jats:italic>. Then (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200001523_inline1\"\/>) is weakly categorical in each<jats:italic>\u03bb<\/jats:italic>\u2265 min<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200001523_inline2\"\/>. If the class (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200001523_inline1\"\/>) is also<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200001523_inline3\"\/>-tame, weak<jats:italic>\u03ba<\/jats:italic>-categoricity is equivalent with<jats:italic>\u03ba<\/jats:italic>-categoricity in the usual sense.<\/jats:p><jats:p>We also discuss the relation between finitary AECs and some other non-elementary frameworks and give several examples.<\/jats:p>","DOI":"10.2178\/jsl\/1309952520","type":"journal-article","created":{"date-parts":[[2011,7,6]],"date-time":"2011-07-06T11:42:35Z","timestamp":1309952555000},"page":"759-806","source":"Crossref","is-referenced-by-count":9,"title":["Categoricity transfer in simple finitary abstract elementary classes"],"prefix":"10.1017","volume":"76","author":[{"given":"Tapani","family":"Hyttinen","sequence":"first","affiliation":[]},{"given":"Meeri","family":"Kes\u00e4l\u00e4","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200001523_ref010","first-page":"1469","volume":"67","author":"Hyttinen","year":"2002","journal-title":"A rank for the class of elementary submodels of a superstable homogeneous model"},{"key":"S0022481200001523_ref008","doi-asserted-by":"crossref","unstructured":"Hyttinen Tapani and Kes\u00e4l\u00e4 Meeri , Finitary abstract elementary classes, Ph.D. thesis, University of Helsinki, Department of Mathematics and Statistics, 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