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We investigate limitwise monotonic numberings. A numbering \u03bd of a family  is limitwise monotonic (l.m.) if every set  is the range of a limitwise monotonic function, uniformly in <jats:italic>k<\/jats:italic>. The set of all l.m. numberings of <jats:italic>S<\/jats:italic> induces the Rogers semilattice . The semilattices  exhibit a peculiar behavior, which puts them <jats:italic>in\u2010between<\/jats:italic> the classical Rogers semilattices (for computable families) and Rogers semilattices of \u2010computable families. We show that every Rogers semilattice of a \u2010computable family is isomorphic to some semilattice . On the other hand, there are infinitely many isomorphism types of classical Rogers semilattices which can be realized as semilattices . In particular, there is an l.m. family <jats:italic>S<\/jats:italic> such that  is isomorphic to the upper semilattice of c.e. <jats:italic>m<\/jats:italic>\u2010degrees. We prove that if an l.m. family <jats:italic>S<\/jats:italic> contains more than one element, then the poset  is infinite, and it is not a lattice. The l.m. numberings form an ideal (w.r.t. reducibility between numberings) inside the class of all \u2010computable numberings. We prove that inside this class, the index set of l.m. numberings is \u2010complete.<\/jats:p>","DOI":"10.1002\/malq.202100077","type":"journal-article","created":{"date-parts":[[2022,3,23]],"date-time":"2022-03-23T02:50:08Z","timestamp":1648003808000},"page":"213-226","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Rogers semilattices of limitwise monotonic numberings"],"prefix":"10.1002","volume":"68","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5834-2770","authenticated-orcid":false,"given":"Nikolay","family":"Bazhenov","sequence":"first","affiliation":[{"name":"Sobolev Institute of Mathematics 4 Acad. Koptyug Ave. Novosibirsk 630090 Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8259-4441","authenticated-orcid":false,"given":"Manat","family":"Mustafa","sequence":"additional","affiliation":[{"name":"Department of Mathematics School of Sciences and Humanities Nazarbayev University 53 Qabanbay Batyr Avenue Nur\u2010Sultan 010000 Kazakhstan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhansaya","family":"Tleuliyeva","sequence":"additional","affiliation":[{"name":"Department of Mathematics School of Sciences and Humanities Nazarbayev University 53 Qabanbay Batyr Avenue Nur\u2010Sultan 010000 Kazakhstan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2022,3,22]]},"reference":[{"key":"e_1_2_8_2_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-68546-5_2"},{"key":"e_1_2_8_3_1","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/257\/04025"},{"key":"e_1_2_8_4_1","doi-asserted-by":"publisher","DOI":"10.1023\/A:1012516217265"},{"key":"e_1_2_8_5_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10469-005-0016-x"},{"key":"e_1_2_8_6_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10469-006-0034-3"},{"key":"e_1_2_8_7_1","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1243948330"},{"key":"e_1_2_8_8_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10469-020-09613-9"},{"key":"e_1_2_8_9_1","doi-asserted-by":"publisher","DOI":"10.1023\/A:1006031314563"},{"key":"e_1_2_8_10_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02219444"},{"key":"e_1_2_8_11_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10469-019-09540-4"},{"key":"e_1_2_8_12_1","doi-asserted-by":"crossref","unstructured":"R. G.Downey A. M.Kach andD.Turetsky Limitwise monotonic functions and their applications in:Proceedings of the 11th Asian Logic Conference edited byT.Arai Q.Feng B.Kim G.Wu andY.Yang(World Scientific Singapore 2011) pp.59\u201385.","DOI":"10.1142\/9789814360548_0004"},{"key":"e_1_2_8_13_1","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19730191901"},{"key":"e_1_2_8_14_1","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19750210164"},{"key":"e_1_2_8_15_1","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19770231902"},{"key":"e_1_2_8_16_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01040653"},{"key":"e_1_2_8_17_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02219785"},{"key":"e_1_2_8_18_1","volume-title":"Theory of Numberings","author":"Ershov Y. 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