{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T00:12:53Z","timestamp":1782259973129,"version":"3.54.5"},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":2202,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2008,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>For a model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004680_inline1\"\/> of Peano Arithmetic, let Lt(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004680_inline1\"\/>) be the lattice of its elementary substructures, and let Lt<jats:sup>+<\/jats:sup> (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004680_inline1\"\/>) be the equivalenced lattice (Lt(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004680_inline1\"\/>),\u2245<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004680_inline1\"\/>), where \u2245<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004680_inline1\"\/> is the equivalence relation of isomorphism on Lt(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004680_inline1\"\/>). It is known that Lt<jats:sup>+<\/jats:sup>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004680_inline1\"\/>) is always a <jats:italic>reasonable<\/jats:italic> equivalenced lattice.<\/jats:p><jats:p>Theorem. <jats:italic>Let L be a finite distributive lattice and let (L, E) be reasonable. If <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004680_inline1\"\/><jats:sub>0<\/jats:sub> is a nonstandard prime model of<\/jats:italic> PA, <jats:italic>then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004680_inline1\"\/><jats:sub>0<\/jats:sub> has a cofinal extension <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004680_inline1\"\/> such that<\/jats:italic> Lt<jats:sup>+<\/jats:sup>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004680_inline1\"\/>) \u2245 (<jats:italic>L,E<\/jats:italic>).<\/jats:p><jats:p>A general method for proving such theorems is developed which, hopefully, will be able to be applied to some nondistributive lattices.<\/jats:p>","DOI":"10.2178\/jsl\/1208358749","type":"journal-article","created":{"date-parts":[[2008,10,14]],"date-time":"2008-10-14T15:19:02Z","timestamp":1223997542000},"page":"193-211","source":"Crossref","is-referenced-by-count":2,"title":["Nondiversity in substructures"],"prefix":"10.1017","volume":"73","author":[{"given":"James H.","family":"Schmerl","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200004680_ref012","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(85)90096-2"},{"key":"S0022481200004680_ref011","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/361\/06593"},{"key":"S0022481200004680_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(76)90015-0"},{"key":"S0022481200004680_ref006","volume-title":"A shorter model theory","author":"Hodges","year":"1997"},{"key":"S0022481200004680_ref002","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-25.4.249"},{"key":"S0022481200004680_ref001","first-page":"572","volume":"43","author":"Abramson","year":"1978","journal-title":"Models without indiscernibles"},{"key":"S0022481200004680_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)70465-8"},{"key":"S0022481200004680_ref004","volume-title":"Ramsey theorey","author":"Graham","year":"1990"},{"key":"S0022481200004680_ref005","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511551574"},{"key":"S0022481200004680_ref007","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198568278.001.0001"},{"key":"S0022481200004680_ref010","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1993-1112501-8"},{"key":"S0022481200004680_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1956-10036-0"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200004680","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,1]],"date-time":"2019-05-01T00:24:48Z","timestamp":1556670288000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200004680\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,3]]},"references-count":12,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2008,3]]}},"alternative-id":["S0022481200004680"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1208358749","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,3]]}}}