{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T00:12:54Z","timestamp":1782259974948,"version":"3.54.5"},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":1197,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2010,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Bounded lattices (that is lattices that are both lower bounded and upper bounded) form a large class of lattices that include all distributive lattices, many nondistributive finite lattices such as the pentagon lattice <jats:italic>N<\/jats:italic><jats:sub>5<\/jats:sub>. and all lattices in any variety generated by a finite bounded lattice. Extending a theorem of Paris for distributive lattices, we prove that if <jats:italic>L<\/jats:italic> is an \u2135<jats:sub>0<\/jats:sub>-algebraic bounded lattice, then every countable nonstandard model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002292_inline1\"\/> of Peano Arithmetic has a cofinal elementary extension <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002292_inline2\"\/> such that the interstructure lattice Lt(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002292_inline2\"\/>\/<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002292_inline1\"\/>) is isomorphic to <jats:italic>L<\/jats:italic>.<\/jats:p>","DOI":"10.2178\/jsl\/1286198152","type":"journal-article","created":{"date-parts":[[2010,10,4]],"date-time":"2010-10-04T13:16:13Z","timestamp":1286198173000},"page":"1366-1382","source":"Crossref","is-referenced-by-count":3,"title":["Infinite substructure lattices of models of Peano Arithmetic"],"prefix":"10.1017","volume":"75","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":"S0022481200002292_ref014","first-page":"203","article-title":"Representations of finite lattices by orders on finite sets","volume":"28","author":"Sivak","year":"1978","journal-title":"Mathematica Slovaca"},{"key":"S0022481200002292_ref015","doi-asserted-by":"publisher","DOI":"10.4064\/fm-95-3-223-237"},{"key":"S0022481200002292_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0059548"},{"key":"S0022481200002292_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/s10469-005-0027-7"},{"key":"S0022481200002292_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1993-1112501-8"},{"key":"S0022481200002292_ref001","doi-asserted-by":"publisher","DOI":"10.4153\/CMB-1970-051-0"},{"key":"S0022481200002292_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF02034334"},{"key":"S0022481200002292_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/042"},{"key":"S0022481200002292_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(79)90007-X"},{"key":"S0022481200002292_ref002","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1979-008-x"},{"key":"S0022481200002292_ref005","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198568278.001.0001"},{"key":"S0022481200002292_ref007","unstructured":"Nation J.B. , e-mail to the author, 06 8, 2009."},{"key":"S0022481200002292_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BF02482893"},{"key":"S0022481200002292_ref009","first-page":"301","volume-title":"Lattice Theory (Proc. 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