{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T09:01:10Z","timestamp":1649062870651},"reference-count":42,"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":1472,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2010,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002851_inline1\" \/>. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable, locally finite lattice) is isomorphic to an initial segment of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002851_inline1\" \/>. Corollaries include the decidability of the two quantifier theory of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002851_inline1\" \/>, and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of \u03c9<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\"><jats:italic>ck<\/jats:italic><\/jats:sup>. Somewhat surprisingly, the set theoretic analog\nof this forcing does not preserve \u03c9<jats:sub>1<\/jats:sub>. On the other hand, we construct countable lattices that are not isomorphic to any initial segment of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002851_inline1\" \/>.<\/jats:p>","DOI":"10.2178\/jsl\/1264433911","type":"journal-article","created":{"date-parts":[[2010,1,25]],"date-time":"2010-01-25T15:38:59Z","timestamp":1264433939000},"page":"103-130","source":"Crossref","is-referenced-by-count":2,"title":["Lattice initial segments of the hyperdegrees"],"prefix":"10.1017","volume":"75","author":[{"given":"Richard A.","family":"Shore","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bj\u00f8rn","family":"Kjos-Hanssen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200002851_ref010","doi-asserted-by":"publisher","DOI":"10.4064\/fm-56-3-325-345"},{"key":"S0022481200002851_ref008","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-28.2.193"},{"key":"S0022481200002851_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0067640"},{"key":"S0022481200002851_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02772668"},{"key":"S0022481200002851_ref020","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19680143002"},{"key":"S0022481200002851_ref030","volume-title":"Proceedings of the symposium on pure mathematics","volume":"XII","author":"Sacks","year":"1971"},{"key":"S0022481200002851_ref005","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19780241903"},{"key":"S0022481200002851_ref028","first-page":"724","volume":"48","author":"Odifreddi","year":"1983","journal-title":"Forcing and reducibilities IT. forcing in fragments of analysis"},{"key":"S0022481200002851_ref009","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19840302605"},{"key":"S0022481200002851_ref007","unstructured":"Dorais F. 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[2002], Lattice initial segments of the Turing degrees, Ph.D. thesis, University of California, Berkeley."},{"key":"S0022481200002851_ref019","doi-asserted-by":"publisher","DOI":"10.2307\/1969708"},{"key":"S0022481200002851_ref024","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(87)90014-5"},{"key":"S0022481200002851_ref011","doi-asserted-by":"publisher","DOI":"10.4064\/fm-61-2-215-223"},{"key":"S0022481200002851_ref023","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21755-9"},{"key":"S0022481200002851_ref025","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-03-03406-8"},{"key":"S0022481200002851_ref027","first-page":"288","volume":"48","author":"Odifreddi","year":"1983","journal-title":"Forcing and reducibilities"},{"key":"S0022481200002851_ref029","volume-title":"Degrees of unsolvability","volume":"55","author":"Sacks","year":"1963"},{"key":"S0022481200002851_ref031","volume-title":"Mathematical Reviews","author":"Sacks","year":"1972"},{"key":"S0022481200002851_ref032","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-12013-2"},{"key":"S0022481200002851_ref033","first-page":"863","article-title":"Algorithmic complexity of algebraic systems","volume":"44","author":"Selivanov","year":"1988","journal-title":"Matematicheskie Zametki"},{"key":"S0022481200002851_ref036","doi-asserted-by":"publisher","DOI":"10.2178\/bsl\/1185803806"},{"key":"S0022481200002851_ref041","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1969-013-2"},{"key":"S0022481200002851_ref037","volume-title":"Computational prospects of infinity, Part II: Presented talks","volume":"15","author":"Shore","year":"2008"},{"key":"S0022481200002851_ref038","unstructured":"Simpson M. 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