{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T03:01:05Z","timestamp":1760151665347,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,4,8]],"date-time":"2022-04-08T00:00:00Z","timestamp":1649376000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62081240416"],"award-info":[{"award-number":["62081240416"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Cyclic associative groupoids (CA-groupoids) and Type-2 cyclic associative groupoids (T2CA-groupoids) are two types of non-associative groupoids which satisfy cyclic associative law and type-2 cyclic associative law, respectively. In this paper, we prove two theorems that weak cancellativity is cancellativity and right quasi-cancellativity is left quasi-cancellativity in a CA-groupoid, thus successfully solving two open problems. Moreover, the relationships among separativity, quasi-cancellativity and commutativity in a CA-groupoid are discussed. Finally, we study the various cancellativities of T2CA-groupoids such as power cancellativity, quasi-cancellativity and cancellativity. By determining the relationships between them, we can illuminate the structure of T2CA-groupoids.<\/jats:p>","DOI":"10.3390\/axioms11040169","type":"journal-article","created":{"date-parts":[[2022,4,8]],"date-time":"2022-04-08T12:11:14Z","timestamp":1649419874000},"page":"169","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Two Open Problems on CA-Groupoids and Cancellativities of T2CA-Groupoids"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9692-0658","authenticated-orcid":false,"given":"Xiaogang","family":"An","sequence":"first","affiliation":[{"name":"School of Mathematics & Data Science, Shaanxi University of Science & Technology, Xi\u2019an 710021, China"}]},{"given":"Xiaohong","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics & Data Science, Shaanxi University of Science & Technology, Xi\u2019an 710021, China"},{"name":"Shaanxi Joint Laboratory of Artificial Intelligence, Shaanxi University of Science & Technology, Xi\u2019an 710021, China"}]},{"given":"Zhirou","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Science, East China University of Technology, Nanchang 330013, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,8]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"On almost semigroups","volume":"2","author":"Kazim","year":"1972","journal-title":"Aligarh Bull. 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