{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T13:14:05Z","timestamp":1700572445247},"reference-count":7,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2010,5,19]],"date-time":"2010-05-19T00:00:00Z","timestamp":1274227200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2010,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Given a (nontrivial) \u0394<jats:sub>1<\/jats:sub> ultrapower \u2131\/\ud835\udcb0, let \u2112\ufe01<jats:sub>U<\/jats:sub> denote the set of all \u03a0<jats:sub>2<\/jats:sub>\u2010correct substructures of \u2131\/\ud835\udcb0; i.e., \u2112\ufe01<jats:sub>U<\/jats:sub> is the collection of all those subsets of |\u2131\/\ud835\udcb0| that are closed under computable (in the sense of \u2131\/\ud835\udcb0) functions. Defining in the obvious way the <jats:italic>lattice<\/jats:italic> \u2112\ufe01(\u2131\/\ud835\udcb0)) with domain \u2112\ufe01<jats:sub>U<\/jats:sub>, we obtain some preliminary results about lattice embeddings into \u2013 or realization as \u2013 an \u2112\ufe01(\u2131\/\ud835\udcb0). The basis for these results, as far as we take the matter, consists of (1) the well\u2010known class of (non\u2010trivial) <jats:italic>minimal<\/jats:italic> \u2131\/\ud835\udcb0's, which function as atoms, and (2) the class of <jats:italic>minimalfree<\/jats:italic> \u2131\/\ud835\udcb0's, to whose nonemptiness a substantial section of the paper is devoted. It is shown that an infinite, convergent monotone sequence together with its limit point is embeddable in an \u2112\ufe01(\u2131\/\ud835\udcb0), and that the initial segment lattices {0, 1,\u2026, <jats:italic>n<\/jats:italic> } are not just embeddable in (as is trivial), but in fact <jats:italic>realizable<\/jats:italic> as, lattices \u2112\ufe01(\u2131\/\ud835\udcb0). Finally, the diamond is (easily) <jats:italic>embeddable<\/jats:italic>; and if it is not realizable, then either the 1 \u2010 3 \u2010 1 lattice or the pentagon is at least embeddable (\u00a9 2010 WILEY\u2010VCH Verlag GmbH &amp; Co. KGaA, Weinheim)<\/jats:p>","DOI":"10.1002\/malq.200810052","type":"journal-article","created":{"date-parts":[[2010,5,20]],"date-time":"2010-05-20T08:24:28Z","timestamp":1274343868000},"page":"323-330","source":"Crossref","is-referenced-by-count":0,"title":["Some observations on the substructure lattice of a \u0394<sub>1<\/sub> ultrapower"],"prefix":"10.1002","volume":"56","author":[{"given":"Thomas G.","family":"McLaughlin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2010,5,19]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"crossref","unstructured":"W. W.Comfort andS.Negrepontis The Theory of Ultrafilters (Springer\u2010Verlag 1974).","DOI":"10.1007\/978-3-642-65780-1"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90002-4"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757881"},{"key":"e_1_2_1_5_2","doi-asserted-by":"crossref","unstructured":"J.Hirschfeld andW.Wheeler Forcing Arithmetic Division Rings. Lecture Notes in Math. #454 (Springer\u2010Verlag 1975).","DOI":"10.1007\/BFb0064082"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(90)90064-9"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(79)90007-X"},{"key":"e_1_2_1_8_2","doi-asserted-by":"crossref","unstructured":"J. B.Paris On Models of Arithmetic. In: Lecture Notes in Math. #255 pp. 251\u2013280 (Springer\u2010Verlag 1972).","DOI":"10.1007\/BFb0059548"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.200810052","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.200810052","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T12:56:40Z","timestamp":1700571400000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.200810052"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,5,19]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2010,6]]}},"alternative-id":["10.1002\/malq.200810052"],"URL":"https:\/\/doi.org\/10.1002\/malq.200810052","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,5,19]]}}}