{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:43:51Z","timestamp":1760237031869,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,22]],"date-time":"2020-02-22T00:00:00Z","timestamp":1582329600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Cyclic associativity can be regarded as a kind of variation symmetry, and cyclic associative groupoid (CA-groupoid) is a generalization of commutative semigroup. In this paper, the various cancellation properties of CA-groupoids, including cancellation, quasi-cancellation and power cancellation, are studied. The relationships among cancellative CA-groupoids, quasi-cancellative CA-groupoids and power cancellative CA-groupoids are found out. Moreover, the concept of variant CA-groupoid is proposed firstly, some examples are presented. It is shown that the structure of variant CA-groupoid is very interesting, and the construction methods and decomposition theorem of variant CA-groupoids are established. <\/jats:p>","DOI":"10.3390\/sym12020315","type":"journal-article","created":{"date-parts":[[2020,2,26]],"date-time":"2020-02-26T04:18:29Z","timestamp":1582690709000},"page":"315","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Some Results on Various Cancellative CA-Groupoids and Variant CA-Groupoids"],"prefix":"10.3390","volume":"12","author":[{"given":"Zhirou","family":"Ma","sequence":"first","affiliation":[{"name":"Department of Mathematics, Shaanxi University of Science &amp; Technology, Xi\u2019an 710021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaohong","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shaanxi University of Science &amp; Technology, Xi\u2019an 710021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5560-5926","authenticated-orcid":false,"given":"Florentin","family":"Smarandache","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,22]]},"reference":[{"key":"ref_1","first-page":"19","article-title":"Cyclic associative groupoids (CA-groupoids) and cyclic associative neutrosophic extended triplet groupoids (CA-NET-groupoids)","volume":"29","author":"Zhang","year":"2019","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Yuan, W.T., and Zhang, X.H. 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