{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T11:55:51Z","timestamp":1759146951378},"reference-count":13,"publisher":"Wiley","issue":"4-5","license":[{"start":{"date-parts":[[2012,7,12]],"date-time":"2012-07-12T00:00:00Z","timestamp":1342051200000},"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":[[2012,8]]},"abstract":"Abstract<\/jats:title>Let a<\/jats:italic> be a Kleene's ordinal notation of a nonzero computable ordinal. We give a sufficient condition on a<\/jats:italic>, so that for every \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\Sigma ^{-1}_a$\\end{document}<\/jats:styled-content>\u2010computable family of two embedded sets, i.e., two sets A<\/jats:italic>, B<\/jats:italic>, with A<\/jats:italic> properly contained in B<\/jats:italic>, the Rogers semilattice of the family is infinite. This condition is satisfied by every notation of \u03c9; moreover every nonzero computable ordinal that is not sum of any two smaller ordinals has a notation that satisfies this condition. On the other hand, we also give a sufficient condition on a<\/jats:italic>, that yields that there is a \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\Sigma ^{-1}_a$\\end{document}<\/jats:styled-content>\u2010computable family of two embedded sets, whose Rogers semilattice consists of exactly one element; this condition is satisfied by all notations of every successor ordinal bigger than 1, and by all notations of the ordinal \u03c9 + \u03c9; moreover every computable ordinal that is sum of two smaller ordinals has a notation that satisfies this condition. We also show that for every nonzero n<\/jats:italic> \u2208 \u03c9, or n<\/jats:italic> = \u03c9, and every notation a<\/jats:italic> of a nonzero ordinal there exists a \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\Sigma ^{-1}_a$\\end{document}<\/jats:styled-content>\u2010computable family of cardinality n<\/jats:italic>, whose Rogers semilattice consists of exactly one element.<\/jats:p>","DOI":"10.1002\/malq.201100114","type":"journal-article","created":{"date-parts":[[2012,8,3]],"date-time":"2012-08-03T18:27:13Z","timestamp":1344018433000},"page":"366-376","source":"Crossref","is-referenced-by-count":6,"title":["Rogers semilattices of families of two embedded sets in the Ershov hierarchy"],"prefix":"10.1002","volume":"58","author":[{"given":"Serikzhan A.","family":"Badaev","sequence":"first","affiliation":[]},{"given":"Mustafa","family":"Manat","sequence":"additional","affiliation":[]},{"given":"Andrea","family":"Sorbi","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,7,12]]},"reference":[{"key":"e_1_2_4_2_1","unstructured":"P. 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L.","year":"1970","journal-title":"III, Algebra Log"},{"key":"e_1_2_4_10_1","volume-title":"Teoriya numeratsi\u01d0, Matematicheskaya Logika i Osnovaniya Matematiki","author":"Ershov Yu. L.","year":"1977"},{"key":"e_1_2_4_11_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02671553"},{"key":"e_1_2_4_12_1","volume-title":"Set Theory","author":"Jech T.","year":"1978"},{"key":"e_1_2_4_13_1","first-page":"311","volume-title":"Computable family of \u03a3\u22121a\u2010sets without Friedberg numberings, in: 6th Conference on Computability in Europe, CiE 2010, Ponta Delgada, Azores, Portugal, June\/July 2010. Abstract and Handout Booklet","author":"Ospichev S.","year":"2010"},{"key":"e_1_2_4_14_1","volume-title":"Theory of Recursive Functions and Effective Computability","author":"Rogers H.","year":"1967"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.201100114","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.201100114","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,31]],"date-time":"2023-08-31T03:39:37Z","timestamp":1693453177000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.201100114"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,7,12]]},"references-count":13,"journal-issue":{"issue":"4-5","published-print":{"date-parts":[[2012,8]]}},"alternative-id":["10.1002\/malq.201100114"],"URL":"https:\/\/doi.org\/10.1002\/malq.201100114","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,7,12]]}}}