{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T15:12:47Z","timestamp":1649171567856},"reference-count":10,"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":14072,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,9]]},"abstract":"<jats:p>Much effort has been spent to prove that the reduced product operation preserves, and sometimes even strengthens the <jats:italic>L<\/jats:italic>-equivalence of structures, where <jats:italic>L<\/jats:italic> is some infinitary language. A similar result is suggested by the following well-known fact:<\/jats:p><jats:p>Assume <jats:italic>D<\/jats:italic> is a nonprincipal ultrafilter on <jats:italic>\u03c9<\/jats:italic> and, for <jats:italic>n<\/jats:italic> \u0404 <jats:italic>\u03c9<\/jats:italic>, <jats:italic>C<jats:sub>n<\/jats:sub><\/jats:italic> is a set. If the ultra-product \u03a0<jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub><jats:italic>C<jats:sub>n<\/jats:sub><\/jats:italic>\/<jats:italic>D<\/jats:italic> is infinite, it has a cardinality \u2265 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053020_inline1\" \/> Hence, by \u0141os' theorem, (i) if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053020_inline2\" \/> and <jats:italic>\u03c6<\/jats:italic>(<jats:italic>x<\/jats:italic>) is a first-order formula, then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053020_inline3\" \/> iff <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053020_inline4\" \/><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053020_inline5\" \/> where <jats:italic>Q<\/jats:italic> is the unary quantifier \u201cthere are <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053020_inline1\" \/> many.\u201d<\/jats:p><jats:p>We shall prove some generalizations of (i). In particular, we show<\/jats:p><jats:p>(ii) if <jats:italic>D<\/jats:italic> is a nonprincipal ultrafilter over <jats:italic>I<\/jats:italic> = \u03c9, and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053020_inline6\" \/> then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053020_inline7\" \/><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053020_inline8\" \/> where <jats:italic>L(Q)<\/jats:italic> is the language obtained from the first-order language by adding the quantifier <jats:italic>Q<\/jats:italic>.<\/jats:p><jats:p>(ii) remains true, if <jats:italic>D<\/jats:italic> is an \u03c9-regular or an atomless filter over a set <jats:italic>I<\/jats:italic>.<\/jats:p><jats:p>Lipner [7] proved that if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053020_inline1\" \/> is regular, then the <jats:italic>L(Q)<\/jats:italic>-equivalence is preserved under direct products. We show that the assumption \u201c<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053020_inline1\" \/> is regular\u201d is necessary.<\/jats:p>","DOI":"10.2307\/2272165","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:36:35Z","timestamp":1146951395000},"page":"410-418","source":"Crossref","is-referenced-by-count":0,"title":["<i>L(Q)<\/i>-preservation theorems"],"prefix":"10.1017","volume":"40","author":[{"given":"J\u00f6rg","family":"Flum","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200053020_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BF02790755"},{"key":"S0022481200053020_ref007","unstructured":"Lipner L. D. , Some aspects of generalized quantifiers, Thesis, University of California, Berkeley, 1970."},{"key":"S0022481200053020_ref002","doi-asserted-by":"publisher","DOI":"10.4064\/fm-49-2-129-141"},{"key":"S0022481200053020_ref010","first-page":"337","article-title":"Generalized products for Q-languages","volume":"17","author":"Wojciechowska","year":"1969","journal-title":"Bulletin de l'Acad\u00e9mie Polonaise des Sciences"},{"key":"S0022481200053020_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BF01237118"},{"key":"S0022481200053020_ref001","first-page":"424","volume":"34","author":"Benda","year":"1969","journal-title":"Reduced products and non-standard logics"},{"key":"S0022481200053020_ref003","doi-asserted-by":"publisher","DOI":"10.4064\/fm-47-1-57-103"},{"key":"S0022481200053020_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0079685"},{"key":"S0022481200053020_ref008","volume-title":"Skrifter utgit av Videnskapsselskapet i Kristiania, I. Klasse","author":"Skolem","year":"1919"},{"key":"S0022481200053020_ref004","article-title":"Generalized quantifiers and reduced products","volume":"21","author":"Flum","year":"1974","journal-title":"Notices of the American Mathematical Society"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200053020","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T19:21:17Z","timestamp":1559157677000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200053020\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,9]]},"references-count":10,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1975,9]]}},"alternative-id":["S0022481200053020"],"URL":"https:\/\/doi.org\/10.2307\/2272165","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,9]]}}}