{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,27]],"date-time":"2023-10-27T08:13:49Z","timestamp":1698394429435},"reference-count":15,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2006,11,13]],"date-time":"2006-11-13T00:00:00Z","timestamp":1163376000000},"content-version":"vor","delay-in-days":4334,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[1995,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A well\u2010known theorem by Martin asserts that the degrees of maximal sets are precisely the high recursively enumerable (r. e.) degrees, and the same is true with \u2018maximal\u2019 replaced by \u2018dense simple\u2019, \u2018<jats:italic>r<\/jats:italic>\u2010maximal\u2019, \u2018strongly hypersimple\u2019 or \u2018finitely strongly hypersimple\u2019. Many other constructions can also be carried out in any given high r. e. degree, for instance <jats:italic>r<\/jats:italic>\u2010maximal or hyperhypersimple sets without maximal supersets (Lerman, Lachlan). In this paper questions of this type are considered systematically. Ultimately it is shown that every conjunction of simplicity\u2010 and non\u2010extensibility properties can be accomplished, unless it is ruled out by well\u2010known, elementary results. Moreover, each construction can be carried out in any given high r. e. degree, as might be expected. For instance, every high r. e. degree contains a dense simple, strongly hypersimple set <jats:italic>A<\/jats:italic> which is contained neither in a hyperhypersimple nor in an <jats:italic>r<\/jats:italic>\u2010maximal set. The paper also contains some auxiliary results, for instance: every r. e. set <jats:italic>B<\/jats:italic> can be transformed into an r. e. set <jats:italic>A<\/jats:italic> such that (i) <jats:italic>A<\/jats:italic> has no dense simple superset, (ii) the transformation preserves simplicity\u2010 or non\u2010extensibility properties as far as this is consistent with (i), and (iii) <jats:italic>A<\/jats:italic> \uf8fd<jats:sub>T<\/jats:sub> <jats:italic>B<\/jats:italic> if <jats:italic>B<\/jats:italic> is high, and <jats:italic>A<\/jats:italic> \u2265<jats:sub>T<\/jats:sub> <jats:italic>B<\/jats:italic> otherwise. Several proofs involve refinements of known constructions; relationships to earlier results are discussed in detail.<\/jats:p>","DOI":"10.1002\/malq.19950410311","type":"journal-article","created":{"date-parts":[[2007,6,2]],"date-time":"2007-06-02T20:54:56Z","timestamp":1180817696000},"page":"395-430","source":"Crossref","is-referenced-by-count":0,"title":["Some New Lattice Constructions in High R. E. Degrees"],"prefix":"10.1002","volume":"41","author":[{"given":"Heinrich","family":"Rolletschek","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,11,13]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.2307\/2032160"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.2307\/2964290"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1972.40.605"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.2307\/2270328"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1968-0227009-1"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.2307\/2270255"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.2307\/2271305"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19660120125"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.2307\/2271653"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19680142105"},{"key":"e_1_2_1_12_2","volume-title":"Theory of Recursive Functions and Effective Computability","author":"Rogers H.","year":"1967"},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.2307\/2273468"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.2307\/2272046"},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02460-7"},{"key":"e_1_2_1_16_2","doi-asserted-by":"publisher","DOI":"10.2307\/2270621"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.19950410311","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.19950410311","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,26]],"date-time":"2023-10-26T14:45:24Z","timestamp":1698331524000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.19950410311"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,1]]},"references-count":15,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1995,1]]}},"alternative-id":["10.1002\/malq.19950410311"],"URL":"https:\/\/doi.org\/10.1002\/malq.19950410311","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,1]]}}}