{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T10:01:25Z","timestamp":1772445685162,"version":"3.50.1"},"reference-count":12,"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":4210,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Assume (\u2135<jats:sub>0<\/jats:sub>, \u2135<jats:sub>1<\/jats:sub>) \u2192 (<jats:italic>\u03bb, \u03bb<\/jats:italic><jats:sup>+<\/jats:sup>). Assume <jats:italic>M<\/jats:italic> is a model of a first order theory <jats:italic>T<\/jats:italic> of cardinality at most <jats:italic>\u03bb<\/jats:italic><jats:sup>+<\/jats:sup> in a language <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120000949X_inline1\"\/> of cardinality \u2264 <jats:italic>\u03bb<\/jats:italic>. Let <jats:italic>N<\/jats:italic> be a model with the same language. Let \u0394 be a set of first order formulas in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120000949X_inline1\"\/> and let <jats:italic>D<\/jats:italic> be a regular filter on \u03bb. Then <jats:italic>M<\/jats:italic> is \u0394-embeddable into the reduced power <jats:italic>N<\/jats:italic><jats:sup>\u03bb<\/jats:sup>\/<jats:italic>D<\/jats:italic>, provided that every \u0394-existential formula true in <jats:italic>M<\/jats:italic> is true also in <jats:italic>N<\/jats:italic>. We obtain the following corollary: for <jats:italic>M<\/jats:italic> as above and <jats:italic>D<\/jats:italic> a regular ultrafilter over <jats:italic>\u03bb<\/jats:italic>, <jats:italic>M<\/jats:italic><jats:sup><jats:italic>\u03bb<\/jats:italic><\/jats:sup>\/<jats:italic>D<\/jats:italic> is <jats:italic>\u03bb<\/jats:italic><jats:sup>++<\/jats:sup>-universal. Our second result is as follows: For <jats:italic>i<\/jats:italic> &lt; <jats:italic>\u03bc<\/jats:italic> let <jats:italic>M<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>, and <jats:italic>N<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>, be elementarily equivalent models of a language which has cardinality \u2264 <jats:italic>\u03bb<\/jats:italic>. Suppose <jats:italic>D<\/jats:italic> is a regular filter on <jats:italic>\u03bb<\/jats:italic> and (\u2135<jats:sub>0<\/jats:sub>, \u2135<jats:sub>1<\/jats:sub>) \u2192 (<jats:italic>\u03bb, \u03bb<\/jats:italic><jats:sup>+<\/jats:sup>) holds. We show that then the second player has a winning strategy in the Ehrenfeucht-Fra\u00efss\u00e9 game of length <jats:italic>\u03bb<\/jats:italic><jats:sup>+<\/jats:sup> on \u03a0<jats:sub><jats:italic>i<\/jats:italic><\/jats:sub><jats:italic>M<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>\/<jats:italic>D<\/jats:italic> and \u03a0<jats:sub><jats:italic>i<\/jats:italic><\/jats:sub><jats:italic>N<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>\/<jats:italic>D<\/jats:italic>. This yields the following corollary: Assume GCH and \u03bb regular (or just (\u2135<jats:sub>0<\/jats:sub>, \u2135<jats:sub>1<\/jats:sub>) \u2192 (<jats:italic>\u03bb, \u03bb<\/jats:italic><jats:sup>+<\/jats:sup>) and 2<jats:sup><jats:italic>\u03bb<\/jats:italic><\/jats:sup> = <jats:italic>\u03bb<\/jats:italic><jats:sup>+<\/jats:sup>. For <jats:italic>L, M<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> and <jats:italic>N<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> be as above, if <jats:italic>D<\/jats:italic> is a regular filter on <jats:italic>\u03bb<\/jats:italic>, then \u03a0<jats:sub><jats:italic>i<\/jats:italic><\/jats:sub><jats:italic>M<\/jats:italic><jats:sup><jats:italic>i<\/jats:italic><\/jats:sup>\/<jats:italic>D<\/jats:italic> \u2245 \u03a0<jats:sub>i<\/jats:sub><jats:italic>N<\/jats:italic><jats:sup><jats:italic>i<\/jats:italic><\/jats:sup>\/<jats:italic>D<\/jats:italic>.<\/jats:p>","DOI":"10.2178\/jsl\/1190150156","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T19:14:26Z","timestamp":1197573266000},"page":"1169-1177","source":"Crossref","is-referenced-by-count":6,"title":["On regular reduced products"],"prefix":"10.1017","volume":"67","author":[{"given":"Juliette","family":"Kennedy","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Saharon","family":"Shelah","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120000949X_ref012","volume-title":"Classification theory and the number of non-isomorphic models","author":"Shelah","year":"1990"},{"key":"S002248120000949X_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/BF02764857"},{"key":"S002248120000949X_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BF02764810"},{"key":"S002248120000949X_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90017-4"},{"key":"S002248120000949X_ref008","unstructured":"Kennedy J. and Shelah S. , On embedding models of arithmetic of cardinality \u21351 into reduced powers, to appear."},{"key":"S002248120000949X_ref007","first-page":"178","volume-title":"Koninkttjke Nederlandse Akademie van Weten-schappen. 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