{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:53:05Z","timestamp":1760147585663,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,18]],"date-time":"2023-02-18T00:00:00Z","timestamp":1676678400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["12271089","11871144"],"award-info":[{"award-number":["12271089","11871144"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>For H a Hopf quasigroup and C, a left quasi H-comodule coalgebra, we show that the smash coproduct C\u22caH (as a symmetry of smash product) is linked to some quotient coalgebra Q=C\/CH*+ by a Morita-Takeuchi context (as a symmetry of Morita context). We use the Morita-Takeuchi setting to prove that for finite dimensional H, equivalent conditions for C\/Q to be a Hopf quasigroup Galois coextension (as a symmetry of Galois extension). In particular, we consider a special case of quasigroup graded coalgebras as an application of our theory.<\/jats:p>","DOI":"10.3390\/sym15020551","type":"journal-article","created":{"date-parts":[[2023,2,20]],"date-time":"2023-02-20T04:58:23Z","timestamp":1676869103000},"page":"551","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Morita-Takeuchi Context and Hopf Coquasigroup Galois Coextensions"],"prefix":"10.3390","volume":"15","author":[{"given":"Huaiwen","family":"Guo","sequence":"first","affiliation":[{"name":"School of Mathematics, Southeast University, Nanjing 210096, China"}]},{"given":"Shuanhong","family":"Wang","sequence":"additional","affiliation":[{"name":"Shing-Tung Yau Center, School of Mathematics, Southeast University, Nanjing 210096, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,18]]},"reference":[{"key":"ref_1","unstructured":"Sweedler, M.E. (1969). 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Mathematics, 11.","DOI":"10.3390\/math11020273"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/551\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:36:07Z","timestamp":1760121367000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/551"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,18]]},"references-count":26,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["sym15020551"],"URL":"https:\/\/doi.org\/10.3390\/sym15020551","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,2,18]]}}}