{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:50:33Z","timestamp":1760028633460},"reference-count":20,"publisher":"Wiley","issue":"1-2","license":[{"start":{"date-parts":[[2016,2,12]],"date-time":"2016-02-12T00:00:00Z","timestamp":1455235200000},"content-version":"vor","delay-in-days":11,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"funder":[{"name":"Science Committee of the Republic of Kazakhstan","award":["3952\/GF4"],"award-info":[{"award-number":["3952\/GF4"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2016,2]]},"abstract":"<jats:p>Using computable reducibility \u2a7d on equivalence relations, we investigate weakly precomplete ceers (a \u201cceer\u201d is a computably enumerable equivalence relation on the natural numbers), and we compare their class with the more restricted class of precomplete ceers. We show that there are infinitely many isomorphism types of universal (in fact uniformly finitely precomplete) weakly precomplete ceers , that are not precomplete; and there are infinitely many isomorphism types of non\u2010universal weakly precomplete ceers. Whereas the Visser space of a precomplete ceer always contains an infinite effectively discrete subset, there exist weakly precomplete ceers whose Visser spaces do not contain infinite effectively discrete subsets. As a consequence, contrary to precomplete ceers which always yield partitions into effectively inseparable sets, we show that although weakly precomplete ceers always yield partitions into computably inseparable sets, nevertheless there are weakly precomplete ceers for which no equivalence class is creative. Finally, we show that the index set of the precomplete ceers is \u03a3<jats:sub>3<\/jats:sub>\u2010complete, whereas the index set of the weakly precomplete ceers is \u03a0<jats:sub>3<\/jats:sub>\u2010complete. As a consequence of these results, we also show that the index sets of the uniformly precomplete ceers and of the e\u2010complete ceers are \u03a3<jats:sub>3<\/jats:sub>\u2010complete.<\/jats:p>","DOI":"10.1002\/malq.201500057","type":"journal-article","created":{"date-parts":[[2016,2,13]],"date-time":"2016-02-13T01:06:29Z","timestamp":1455325589000},"page":"111-127","source":"Crossref","is-referenced-by-count":15,"title":["Weakly precomplete computably enumerable equivalence relations"],"prefix":"10.1002","volume":"62","author":[{"given":"Serikzhan","family":"Badaev","sequence":"first","affiliation":[{"name":"Department of Mechanics and Mathematics Al\u2010Farabi Kazakh National University Almaty 050038 Kazakhstan"}]},{"given":"Andrea","family":"Sorbi","sequence":"additional","affiliation":[{"name":"Dipartimento di Ingegneria dell'Informazione e Scienze Matematiche Universit\u00e0 Degli Studi di Siena 53100 Siena Italy"}]}],"member":"311","published-online":{"date-parts":[[2016,2,12]]},"reference":[{"key":"e_1_2_6_2_1","doi-asserted-by":"publisher","DOI":"10.1017\/jsl.2013.8"},{"key":"e_1_2_6_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF00972779"},{"key":"e_1_2_6_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01837553"},{"key":"e_1_2_6_5_1","doi-asserted-by":"crossref","first-page":"221","DOI":"10.4064\/fm-124-3-221-233","article-title":"Equivalence relations induced by extensional formulae: Classification by means of a new fixed point property","volume":"124","author":"Bernardi C.","year":"1984","journal-title":"Fundam. Math."},{"key":"e_1_2_6_6_1","doi-asserted-by":"publisher","DOI":"10.2307\/2273443"},{"key":"e_1_2_6_7_1","doi-asserted-by":"crossref","first-page":"15","DOI":"10.3233\/COM-2012-004","article-title":"The hierarchy of equivalence relations on the natural numbers","volume":"1","author":"Coskey S.","year":"2012","journal-title":"Computability"},{"key":"e_1_2_6_8_1","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1090\/S0002-9939-1954-0063995-6","article-title":"A theorem on hypersimple sets","volume":"5","author":"Dekker J. C. E.","year":"1954","journal-title":"Proc. Amer. Math. Soc."},{"key":"e_1_2_6_9_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02218645"},{"key":"e_1_2_6_10_1","volume-title":"\u0422\u0435\u043e\u0440\u0438\u044f \u043d\u0443\u043c\u0435\u0440\u0430\u0446\u0438\u0439, \u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u041b\u043e\u0433\u0438\u043a\u0430 \u0438 \u041e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f \u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438","author":"Ershov Y. L."},{"key":"e_1_2_6_11_1","first-page":"473","volume-title":"Studies in Logic and the Foundations of Mathematics Vol. 140","author":"Ershov Y. L.","year":"1999"},{"key":"e_1_2_6_12_1","doi-asserted-by":"publisher","DOI":"10.1023\/A:1010521410739"},{"key":"e_1_2_6_13_1","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19870330106"},{"issue":"2","key":"e_1_2_6_14_1","first-page":"4","article-title":"Sets with complete numberings","volume":"2","author":"Mal'tsev A. I.","year":"1963","journal-title":"Algebra Log."},{"issue":"2","key":"e_1_2_6_15_1","first-page":"278","article-title":"Positive and negative enumerations","volume":"160","author":"Mal'tsev A. I.","year":"1965","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"e_1_2_6_16_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF00284977"},{"key":"e_1_2_6_17_1","volume-title":"Theory of Recursive Functions and Effective Computability","author":"Rogers Jr. H.","year":"1967"},{"key":"e_1_2_6_18_1","unstructured":"L.SanMauro Forma e complessit\u00e0. Uno studio dei gradi delle relazioni di equivalenza ricorsivamente enumerabili Master's thesis University of Siena 2011."},{"key":"e_1_2_6_19_1","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19960420107"},{"key":"e_1_2_6_20_1","volume-title":"A study of computable functions and computably generated sets. Perspectives in Mathematical Logic","author":"Soare R. I.","year":"1987"},{"key":"e_1_2_6_21_1","first-page":"259","volume-title":"Lambda Calculus and Formalism","author":"Visser A.","year":"1980"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.201500057","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.201500057","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,3]],"date-time":"2023-10-03T20:25:07Z","timestamp":1696364707000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.201500057"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,2]]},"references-count":20,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2016,2]]}},"alternative-id":["10.1002\/malq.201500057"],"URL":"https:\/\/doi.org\/10.1002\/malq.201500057","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,2]]}}}