{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T18:48:01Z","timestamp":1767898081678,"version":"3.49.0"},"reference-count":32,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/100000001","name":"NSF","doi-asserted-by":"publisher","award":["DMS-2451099"],"award-info":[{"award-number":["DMS-2451099"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"NSF","doi-asserted-by":"publisher","award":["SFI-MPS-TSM-00013602"],"award-info":[{"award-number":["SFI-MPS-TSM-00013602"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,1,8]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>We study the structure of the c.e. $Q$- and $sQ$-degrees. For both structures, we show that there are join irreducible degrees but no Ahmad pairs. We show that the structures are not distributive, and that the lattice $N_{5}$ embeds in both structures. On the other hand the lattice $M_{5}$ cannot be embedded in the c.e. $sQ$-degrees, but a critical triple can be embedded. Finally, we show that no initial segment of the c.e. $sQ$-degrees or the c.e. $Q$-degrees is a lattice.<\/jats:p>","DOI":"10.1093\/logcom\/exaf076","type":"journal-article","created":{"date-parts":[[2025,12,3]],"date-time":"2025-12-03T13:14:00Z","timestamp":1764767640000},"source":"Crossref","is-referenced-by-count":0,"title":["On quasi-reducibility for c.e. sets Part I. The structure of the\n                    <i>Q<\/i>\n                    -degrees and the\n                    <i>sQ<\/i>\n                    -degrees"],"prefix":"10.1093","volume":"36","author":[{"given":"Sapir","family":"Ben-Shahar","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Wisconsin-Madison , 480 Lincoln Drive, Madison, WI 53706,","place":["USA"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rodney G","family":"Downey","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Victoria University of Wellington , 79 Kelburn Parade, 6012, Wellington,","place":["New Zealand"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mariya","family":"Soskova","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Wisconsin-Madison , 480 Lincoln Drive, Madison, WI 53706,","place":["USA"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2026,1,8]]},"reference":[{"key":"2026010808235844200_ref1","volume-title":"Some Results on the Structure of theEnumeration Degrees","author":"Ahmad","year":"1989"},{"key":"2026010808235844200_ref2","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1007\/BF01463347","article-title":"Algebraically closed groups","volume":"13","author":"Belegradek","year":"1974","journal-title":"Algebra Logika"},{"key":"2026010808235844200_ref3","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1090\/S0002-9947-1992-1097164-6","article-title":"Automorphisms of the lattice of recursively enumerable sets: Promptly simple sets","volume":"332","author":"Cholak","year":"1992","journal-title":"Transactions of the American Mathematical Society"},{"key":"2026010808235844200_ref4","first-page":"143","article-title":"$tt$- and $m$-degrees","volume":"12","author":"D\u00ebgtev","year":"1973","journal-title":"Algebra Logika"},{"key":"2026010808235844200_ref5","first-page":"233","volume":"29","author":"Demuth","year":"1988","journal-title":"Commentationes Mathematicae Universitatis Carolinae"},{"key":"2026010808235844200_ref6","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1090\/S0002-9947-1985-0797064-9","article-title":"The degrees of r.e. sets without the universal splitting property","volume":"291","author":"Downey","year":"1985","journal-title":"Transactions of the American Mathematical Society"},{"key":"2026010808235844200_ref7","doi-asserted-by":"crossref","first-page":"299","DOI":"10.2140\/pjm.1987.127.299","article-title":"Subsets of hypersimple sets","volume":"127","author":"Downey","year":"1987","journal-title":"Pacific Journal of Mathematics"},{"key":"2026010808235844200_ref8","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/0168-0072(90)90062-7","article-title":"Lattice nonembeddings and initial segments of the recursively enumerable degrees","volume":"49","author":"Downey","year":"1990","journal-title":"Annals of Pure and Applied Logic"},{"key":"2026010808235844200_ref9","doi-asserted-by":"crossref","DOI":"10.23943\/princeton\/9780691199665.001.0001","article-title":"A hierarchy of Turing degrees, volume 206 of Annals of Mathematics Studies","volume-title":"A Transfinite Hierarchy of Lowness Notions in the Computably Enumerable Degrees, Unifying Classes, and Natural Definability","author":"Downey","year":"2020"},{"key":"2026010808235844200_ref10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0168-0072(98)00013-X","article-title":"Computably enumerable sets and quasi-reducibility","volume":"95","author":"Downey","year":"1998","journal-title":"Annals of Pure and Applied Logic"},{"key":"2026010808235844200_ref11","first-page":"620","volume":"10","author":"Ershov","year":"1971","journal-title":"Positive equivalences. 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