{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T02:51:32Z","timestamp":1777517492680,"version":"3.51.4"},"reference-count":29,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[2021,1,1]],"date-time":"2021-01-01T00:00:00Z","timestamp":1609459200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Computability"],"published-print":{"date-parts":[[2021,1,20]]},"abstract":"<jats:p>We investigate the complexity of embeddings between bi-embeddable structures. In analogy with categoricity spectra, we define the bi-embeddable categoricity spectrum of a structure [Formula: see text] as the family of Turing degrees that compute embeddings between any computable bi-embeddable copies of [Formula: see text]; the degree of bi-embeddable categoricity of [Formula: see text] is the least degree in this spectrum (if it exists). We extend many known results about categoricity spectra to the case of bi-embeddability. In particular, we exhibit structures without degree of bi-embeddable categoricity, and we show that every degree d.c.e above [Formula: see text] for \u03b1 a computable successor ordinal and [Formula: see text] for \u03bb a computable limit ordinal is a degree of bi-embeddable categoricity. We also give examples of families of degrees that are not bi-embeddable categoricity spectra.<\/jats:p>","DOI":"10.3233\/com-190289","type":"journal-article","created":{"date-parts":[[2020,5,22]],"date-time":"2020-05-22T11:21:35Z","timestamp":1590146495000},"page":"1-16","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":7,"title":["Degrees of bi-embeddable categoricity"],"prefix":"10.1177","volume":"10","author":[{"given":"Nikolay","family":"Bazhenov","sequence":"first","affiliation":[{"name":"Sobolev Institute of Mathematics, 4 Acad. Koptyug Ave., Novosibirsk, 630090, Russia"},{"name":"Novosibirsk State University, 2 Pirogova St., Novosibirsk, 630090, Russia."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ekaterina","family":"Fokina","sequence":"additional","affiliation":[{"name":"Institute of Discrete Mathematics, Vienna University of Technology, Wiedner Hauptstrasse 8-10\/104, 1040 Vienna, Austria."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dino","family":"Rossegger","sequence":"additional","affiliation":[{"name":"Department of Pure Mathematics, University of Waterloo, 200 University Ave West, Waterloo, Ontario, Canada."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luca","family":"San Mauro","sequence":"additional","affiliation":[{"name":"Institute of Discrete Mathematics, Vienna University of Technology, Wiedner Hauptstrasse 8-10\/104, 1040 Vienna, Austria."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2021,1,1]]},"reference":[{"key":"ref001","doi-asserted-by":"publisher","DOI":"10.1215\/00294527-3496154"},{"key":"ref002","doi-asserted-by":"publisher","DOI":"10.1016\/s0049-237x(00)80006-3"},{"key":"ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(89)90015-8"},{"key":"ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1986-0860377-7"},{"key":"ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(90)90004-L"},{"key":"ref006","doi-asserted-by":"crossref","unstructured":"N.\u00a0Bazhenov, Linear orders and categoricity spectra, in: Proceedings of the 14th and 15th Asian Logic Conferences, B.\u00a0Kim, J.\u00a0Brendle, G.\u00a0Lee, F.\u00a0Liu, R.\u00a0Ramanujam, S.M.\u00a0Srivastava, A.\u00a0Tsuboi and L.\u00a0Yu, eds, World Scientific, 2019, pp.\u00a035\u201352.","DOI":"10.1142\/9789813237551_0002"},{"key":"ref007","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-018-0650-3"},{"key":"ref008","doi-asserted-by":"publisher","DOI":"10.1007\/s10469-018-9511-8"},{"key":"ref009","doi-asserted-by":"publisher","DOI":"10.1007\/s10469-016-9381-x"},{"key":"ref010","doi-asserted-by":"publisher","DOI":"10.1134\/S1995080217040035"},{"key":"ref011","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1245158098"},{"key":"ref012","doi-asserted-by":"publisher","DOI":"10.1215\/00294527-1960479"},{"key":"ref013","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2014.09.022"},{"key":"ref014","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-009-0160-4"},{"key":"ref015","doi-asserted-by":"publisher","DOI":"10.1002\/malq.201800056"},{"key":"ref016","doi-asserted-by":"publisher","DOI":"10.3233\/COM-140027"},{"key":"ref017","doi-asserted-by":"publisher","DOI":"10.3233\/COM-170078"},{"key":"ref018","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1294170999"},{"key":"ref019","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.1956.0003"},{"key":"ref020","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-07-04285-7"},{"key":"ref021","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1972-0316227-0"},{"key":"ref022","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-05-04005-5"},{"key":"ref023","unstructured":"A.I.\u00a0Maltsev, On recursive Abelian groups, Soviet Mathematics. 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