{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:26:33Z","timestamp":1760149593037,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,21]],"date-time":"2023-08-21T00:00:00Z","timestamp":1692576000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004242","name":"Princess Nourah Bint Abdulrahman University","doi-asserted-by":"publisher","award":["PNURSP2023R231"],"award-info":[{"award-number":["PNURSP2023R231"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The complete characterization of saturated varieties of semigroups remains an unsolved problem. The primary objective of this paper is to make significant progress in this direction. We initially demonstrate that the variety of semigroups defined by the identity axy=ayxa is saturated. The next main result establishes that the variety of semigroups determined by the identity axy=ayax is saturated. Finally, we show that medial semigroups satisfying the identity xy=xyn, where n\u22652, are also saturated. These results collectively lead to the conclusion that epis from these saturated varieties are onto. This paper thus offers substantial progress towards the comprehensive characterization of saturated varieties of semigroups.<\/jats:p>","DOI":"10.3390\/sym15081612","type":"journal-article","created":{"date-parts":[[2023,8,21]],"date-time":"2023-08-21T08:53:31Z","timestamp":1692608011000},"page":"1612","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Saturated Varieties of Semigroups"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8517-4151","authenticated-orcid":false,"given":"Muneer","family":"Nabi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Chandigarh University, Mohali 140413, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7856-2861","authenticated-orcid":false,"given":"Amal S.","family":"Alali","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"given":"Sakeena","family":"Bano","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Central University of Kashmir, Ganderbal 191131, India"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/0021-8693(67)90010-5","article-title":"Epimorphisms and Dominions II","volume":"6","author":"Howie","year":"1967","journal-title":"J. Algebra"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/BF02676649","article-title":"Epis are onto for generalized inverse semigroups","volume":"23","author":"Higgins","year":"1981","journal-title":"Semigroup Forum"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1090\/S0002-9947-1985-0768723-9","article-title":"Epimorphically closed permutative varieties","volume":"287","author":"Khan","year":"1985","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Howie, J.M. (1995). Fundamentals of Semigroup Theory, Clarendon Press.","DOI":"10.1093\/oso\/9780198511946.001.0001"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Isbell, J.R. (1965, January 12). Epimorphisms and Dominions. Proceedings of the Conference on Categorical Algebra, La Jolla, CA, USA.","DOI":"10.1007\/978-3-642-99902-4_9"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1017\/S1446788700023041","article-title":"On saturated Permutative Varieties and Consequences of Permutation Identities","volume":"38","author":"Khan","year":"1985","journal-title":"J. Aust. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1017\/S1446788700024629","article-title":"Saturated and epimorphically closed varieties of semigroups","volume":"36","author":"Higgins","year":"1984","journal-title":"J. Aust. Math. Soc."},{"key":"ref_8","first-page":"110","article-title":"On saturated semigroups","volume":"4","author":"Alam","year":"2019","journal-title":"J. Abstr. Comput. Math."},{"key":"ref_9","first-page":"231","article-title":"On Epimorphisms and Structurally Regular Semigroups","volume":"15","author":"Shah","year":"2021","journal-title":"Categ. Gen. Algebr. Struct. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/s00233-019-10050-z","article-title":"Epimorphisms, Dominions and H-commutative semigroups","volume":"100","author":"Alam","year":"2020","journal-title":"Semigroup Forum"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1080\/00927872.2022.2095565","article-title":"Closed and Saturated Varieties of Semigroups","volume":"15","author":"Ahanger","year":"2023","journal-title":"Commun. Algebra"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Alam, N., Khan, N.M., Obeidat, S., Kirmani, S.A.K., and Ahmed, J. (2022). Epimorphisms, dominions, and various classes of saturated semigroups. J. Math., 2684474.","DOI":"10.1155\/2022\/2684474"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Alali, A.S., Bano, S., and Nabi, M. (2023). Saturated (n,m)-regular semigroups. Mathematics, 11.","DOI":"10.3390\/math11092203"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/8\/1612\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:38:17Z","timestamp":1760128697000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/8\/1612"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,21]]},"references-count":13,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["sym15081612"],"URL":"https:\/\/doi.org\/10.3390\/sym15081612","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,8,21]]}}}