{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T15:45:22Z","timestamp":1767109522349,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2019,6,20]],"date-time":"2019-06-20T00:00:00Z","timestamp":1560988800000},"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>The quasi-cancellativity of Abel Grassmann\u2018s groupoids (AG-groupoids) are discussed and two conjectures are partially solved. First, the following conjecture is proved to be true: every AG-3-band is quasi-cancellative. Moreover, a new notion of AG-(4,1)-band is proposed, and it is also proved that every AG-(4,1)-band is quasi-cancellative. Second, the notions of left (right) quasi-cancellative AG-groupoids and power-cancellative AG-groupoids are proposed, and the following results are obtained: for an AG*-groupoid or AG**-groupoid, it is left quasi-cancellative if and only if it is right quasi-cancellative; for a power-cancellative and locally power-associative AG-groupoid, it is left quasi-cancellative if and only if it is right quasi-cancellative. Finally, a general result is proved, that for any AG-groupoid, if it is left quasi-cancellative then it is right quasi-cancellative.<\/jats:p>","DOI":"10.3390\/sym11060816","type":"journal-article","created":{"date-parts":[[2019,6,20]],"date-time":"2019-06-20T10:49:59Z","timestamp":1561027799000},"page":"816","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On Two Conjectures of Abel Grassmann\u2019s Groupoids"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2320-0884","authenticated-orcid":false,"given":"Xiaohong","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Arts and Sciences, Shaanxi University of Science &amp; Technology, Xi\u2019an 710021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yingcang","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Science, Xi\u2019an Polytechnic University, Xi\u2019an 710048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peng","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Arts and Sciences, Shaanxi University of Science &amp; Technology, Xi\u2019an 710021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,20]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"On almost semigroups","volume":"2","author":"Kazim","year":"1972","journal-title":"Alig. 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