{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:31:32Z","timestamp":1760243492185,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,11]],"date-time":"2022-10-11T00:00:00Z","timestamp":1665446400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Guangdong Province","award":["2021A1515010381","2022KTSCX145"],"award-info":[{"award-number":["2021A1515010381","2022KTSCX145"]}]},{"name":"Innovation Project of Department of Education of Guangdong Province","award":["2021A1515010381","2022KTSCX145"],"award-info":[{"award-number":["2021A1515010381","2022KTSCX145"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we study the complete \u22c6-metric semigroups and groups and the Ra\u01d0kov completion of invariant \u22c6-metric groups. We obtain the following. (1) Let (X,d\u22c6) be a complete \u22c6-metric space containing a semigroup (group) G that is a dense subset of X. If the restriction of d\u22c6 on G is invariant, then X can become a semigroup (group) containing G as a subgroup, and d\u22c6 is invariant on X. (2) Let (G,d\u22c6) be a \u22c6-metric group such that d\u22c6 is invariant on G. Then, (G,d\u22c6) is complete if and only if (G,\u03c4d\u22c6) is Ra\u01d0kov complete.<\/jats:p>","DOI":"10.3390\/axioms11100546","type":"journal-article","created":{"date-parts":[[2022,10,11]],"date-time":"2022-10-11T22:18:13Z","timestamp":1665526693000},"page":"546","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Complete Invariant \u22c6-Metrics on Semigroups and Groups"],"prefix":"10.3390","volume":"11","author":[{"given":"Shi-Yao","family":"He","sequence":"first","affiliation":[{"name":"School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian-Cai","family":"Wei","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5584-2924","authenticated-orcid":false,"given":"Li-Hong","family":"Xie","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1007\/s00233-022-10259-5","article-title":"Various types of completeness in topologized semilattices","volume":"104","author":"Kazachenko","year":"2022","journal-title":"Semigroup Forum"},{"key":"ref_2","unstructured":"Ha, K.Y., and Lee, J.B. (2022). Left invariant Lorentzian metrics and curvatures on non-unimodular Lie groups of dimension three. arXiv."},{"key":"ref_3","first-page":"326","article-title":"Fuzzy metrics and statistical metric spaces","volume":"11","author":"Kramosil","year":"1975","journal-title":"Kybernetika"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/j.fss.2014.06.016","article-title":"Gromov-Hausdorff convergence of non-Archimedean fuzzy metric spaces","volume":"267","author":"Macario","year":"2015","journal-title":"Fuzzy Sets Syst."},{"key":"ref_5","first-page":"925","article-title":"Some results on fuzzy metric spaces","volume":"19","author":"Rano","year":"2011","journal-title":"J. Fuzzy Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/j.topol.2013.10.014","article-title":"Domain-theoretic approach to fuzzy metric spaces","volume":"163","author":"Ricarte","year":"2014","journal-title":"Topol. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/S0165-0114(00)00085-3","article-title":"On fuzzy metric groups","volume":"124","author":"Romaguera","year":"2001","journal-title":"Fuzzy Sets Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/j.fss.2017.05.022","article-title":"Fuzzy quasi-pseudometrics on algebraic structures","volume":"330","author":"Sanchis","year":"2018","journal-title":"Fuzzy Sets Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.fss.2016.12.019","article-title":"Complete invariant fuzzy metrics on groups","volume":"330","author":"Sanchis","year":"2018","journal-title":"Fuzzy Sets Syst."},{"key":"ref_10","first-page":"766","article-title":"Complete invariant fuzzy metrics on semigroups and groups","volume":"11","author":"Tu","year":"2021","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_11","unstructured":"Khatami, S.M.A., and Mirzavaziri, M. (2020). Yet another generalization of the notion of a metric space. arXiv."},{"key":"ref_12","unstructured":"He, S.Y., Jin, Y.Y., and Xie, L.H. (2022). \u22c6-quasi-pseudometrics on algebraic structures. Appl. Math., submitted."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Arhangel\u2019skii, A.V., and Tkachenko, M.G. (2008). Topological Groups and Related Structures, Atlantis Press.","DOI":"10.2991\/978-94-91216-35-0"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/10\/546\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:50:14Z","timestamp":1760143814000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/10\/546"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,11]]},"references-count":13,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["axioms11100546"],"URL":"https:\/\/doi.org\/10.3390\/axioms11100546","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,10,11]]}}}