{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T13:22:45Z","timestamp":1648819365208},"reference-count":14,"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":7041,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1994,12]]},"abstract":"Embeddings into initial segments of (the lattice of ideals of r.e. degrees) are relevant for the investigation of decidability of (fragments of) the theory of (with constant-symbols denoting principal ideals). Interest in this investigation stems from open problems concerning decidability of fragments of the theory of R<\/jats:bold> (the usl of r.e. degrees) and embeddings into R<\/jats:bold>. Ambos-Spies and Lerman established comprehensive nonembeddability [AS\/LR1] and embeddability [AS\/LR2] conditions, but their complementarity is open. (For background on embeddings and decidability consult Lerman [LR1] and Lerman and Shore [LR\/SH].) The first systematic investigation of embeddings into is in Calhoun [CL], where the existence of uniform bounds, in the case of embeddings into , on the number of \u201cjoin traces\u201d arising with nondistributive lattices is shown to remove \u201cobstructions\u201d to lattice-embeddings into R<\/jats:bold> (such as typified in the nonembeddable lattice S<\/jats:italic>8<\/jats:sub>, see [LA\/SR], [AS\/LR1\/). The analysis to follow extends this, by showing that it also removes the \u201cobstruction\u201d to lattice-embeddings into initial segments of R<\/jats:bold>, arising with Lachlan's M<\/jats:italic>5<\/jats:sub>, which embeds into R<\/jats:bold>, but not into every nontrivial initial segment R[0, a]<\/jats:bold>, by a result of Downey [DW]. The techniques are Lachlan's [LA], as adapted in Lerman's \u201cpinball machine model\u201d [LR1, LR2], combined with Fejer's \u201cmeet-trick\u201d [FJ] as used by Downey [DW] in the proof that every distributive lattice is embeddable into every nontrivial initial segment of r.e. degrees R[0, a], a > 0<\/jats:bold>. We present a sufficient condition, \u201callowing evasion\u201d [Definition 4.2] for lattice embeddability into every [0, a], a > 0<\/jats:bold>.<\/jats:p>","DOI":"10.2307\/2275708","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:53:29Z","timestamp":1146956009000},"page":"1326-1350","source":"Crossref","is-referenced-by-count":1,"title":["Initial segments of the lattice of ideals of r.e. degrees"],"prefix":"10.1017","volume":"59","author":[{"given":"Frank P.","family":"Weber","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200019307_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(90)90062-7"},{"key":"S0022481200019307_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(83)90028-3"},{"key":"S0022481200019307_ref012","first-page":"824","volume":"58","author":"Shore","year":"1993","journal-title":"Working below a high recursively enumerable degree"},{"key":"S0022481200019307_ref001","first-page":"257","volume":"51","author":"Ambos-Spies","year":"1986","journal-title":"Lattice embeddings into the recursively enumerable degrees. I"},{"key":"S0022481200019307_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(80)90027-4"},{"key":"S0022481200019307_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21755-9"},{"key":"S0022481200019307_ref009","first-page":"13","volume-title":"[NR\/SH]","author":"Lerman"},{"key":"S0022481200019307_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0059544"},{"key":"S0022481200019307_ref014","volume-title":"Recursively enumerable sets and degrees","author":"Soare","year":"1986"},{"key":"S0022481200019307_ref002","first-page":"735","volume":"54","author":"Ambos-Spies","year":"1989","journal-title":"Lattice embeddings into the recursively enumerable degrees. II"},{"key":"S0022481200019307_ref003","unstructured":"Calhoun W. C. , The lattice of ideals of recursively enumerable degrees, Ph.D. thesis, University of California, Berkeley, California, 1990."},{"key":"S0022481200019307_ref010","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1988-0973174-0"},{"key":"S0022481200019307_ref011","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/042"},{"key":"S0022481200019307_ref013","first-page":"53","volume-title":"[NR\/SH]","author":"Soare"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200019307","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T00:40:48Z","timestamp":1557880848000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200019307\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,12]]},"references-count":14,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1994,12]]}},"alternative-id":["S0022481200019307"],"URL":"https:\/\/doi.org\/10.2307\/2275708","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,12]]}}}