{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:55:29Z","timestamp":1760237729306,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,7,30]],"date-time":"2022-07-30T00:00:00Z","timestamp":1659139200000},"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>Tarski associative groupoid (TA-groupoid) is a kind of non-associative groupoid satisfying Tarski associative law. In this paper, the new notions of transposition regular TA-groupoid are proposed and their properties and structural characteristics are studied by using band and quasi-separativity. In particular, the following conclusions are strictly proved: (1) every left transposition regular TA-groupoid is a semigroup; (2) every left transposition regular TA-groupoid is the disjoint union of sub Abelian groups; and (3) a finite TA-groupoid with quasi-separativity and a finite left transposition regular TA-groupoid are equivalent.<\/jats:p>","DOI":"10.3390\/axioms11080378","type":"journal-article","created":{"date-parts":[[2022,7,31]],"date-time":"2022-07-31T21:49:02Z","timestamp":1659304142000},"page":"378","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Transposition Regular TA-Groupoids and Their Structures"],"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"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1090\/S0002-9947-1929-1501476-0","article-title":"On a generalization of the associative law","volume":"31","author":"Suschkewitsch","year":"1929","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"420","DOI":"10.2307\/1968930","article-title":"Sets of postulates for Boolean groups","volume":"40","author":"Bernstein","year":"1939","journal-title":"Ann. Math."},{"key":"ref_3","first-page":"205","article-title":"Some functional equations related with the associative law","volume":"3","year":"1954","journal-title":"Publ. Math. Debr."},{"key":"ref_4","first-page":"125","article-title":"CM solutions of some functional equations of associative type","volume":"24","author":"Maksa","year":"2004","journal-title":"Annales Univ. Sci. Budapest. Sect. Comp."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/s00010-015-0364-0","article-title":"Power series solutions of Tarski\u2019s associativity law and of the cyclic associativity law","volume":"90","author":"Tomaschek","year":"2016","journal-title":"Aequationes Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1007\/BF01111019","article-title":"Ringe mit x(yz) = (yx)z","volume":"99","author":"Thedy","year":"1967","journal-title":"Math. Z."},{"key":"ref_7","first-page":"187","article-title":"Para-associative groupoids","volume":"18","author":"Pushkashu","year":"2010","journal-title":"Quasigroups Relat. Syst."},{"key":"ref_8","unstructured":"Kandasamy, W.B.V., Smarandache, F., and Chetry, M.K. (2010). Interval Groupoids, Infinite Study."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Zhang, X., Yuan, W., Chen, M., and Smarandache, F. (2020). A kind of variation symmetry: Tarski associative groupoids (TA-groupoids) and Tarski associative neutrosophic extended triplet groupoids (TA-NET-groupoids). Symmetry, 12.","DOI":"10.3390\/sym12050714"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1017\/S030500410003036X","article-title":"A note on inverse semigroups","volume":"51","author":"Munn","year":"1955","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0021-8693(73)90150-6","article-title":"On regular semigroups","volume":"24","author":"Hall","year":"1973","journal-title":"J. Algebra"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/BF02194760","article-title":"The structure of regular semigroups, I: A representation","volume":"8","author":"Grillet","year":"1974","journal-title":"Semigroup Forum"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1007\/BF02194766","article-title":"The structure of regular semigroups, II: Cross-connections","volume":"8","author":"Grillet","year":"1974","journal-title":"Semigroup Forum"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1007\/BF02194767","article-title":"The structure of regular semigroups, III: The reduced case","volume":"8","author":"Grillet","year":"1974","journal-title":"Semigroup Forum"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1007\/s10474-018-0888-6","article-title":"Inductive groupoids and cross-connections of regular semigroups","volume":"157","author":"Muhammed","year":"2019","journal-title":"Acta Math. Hung."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1007\/s11005-015-0760-3","article-title":"Relational symplectic groupoids","volume":"105","author":"Cattaneo","year":"2015","journal-title":"Lett. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Zhang, X., and Du, Y. (2022). Left (right) regular and transposition regular semigroups and their structures. Mathematics, 10.","DOI":"10.3390\/math10071021"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Du, Y., Zhang, X., and An, X. (2022). Transposition regular AG-groupoids and their decomposition theorems. Mathematics, 10.","DOI":"10.3390\/math10091396"},{"key":"ref_19","first-page":"371","article-title":"AG-test and some general properties of Abel-Grassmann\u2019s groupoids","volume":"6","year":"1995","journal-title":"Pure Math. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.9734\/BJMCS\/2016\/21867","article-title":"On cyclic associative Abel-Grassman groupoids","volume":"12","author":"Iqbal","year":"2016","journal-title":"Br. J. Math. Comput. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1016\/0021-8693(86)90003-7","article-title":"A partial order in completely regular semigroups","volume":"98","author":"Drazin","year":"1986","journal-title":"J. Algebra"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1007\/s00233-004-0111-7","article-title":"On quasi-separative semigroups","volume":"70","author":"Krasilnikova","year":"2005","journal-title":"Semigroup Forum"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"An, X., Zhang, X., and Ma, Z. (2022). Two open problems on CA-groupoids and cancellativities of T2CA-groupoids. Axioms, 11.","DOI":"10.3390\/axioms11040169"},{"key":"ref_24","first-page":"95","article-title":"Abel-Grassmann\u2019s bands","volume":"11","year":"2004","journal-title":"Quasigroups Relat. Syst."},{"key":"ref_25","first-page":"175","article-title":"Composition of Abel-Grassmann\u2019s 3-bands","volume":"34","year":"2004","journal-title":"Novi Sad J. Math."},{"key":"ref_26","first-page":"431","article-title":"Band decompositions of Abel-Grassmann\u2019s groupoids","volume":"12","year":"2001","journal-title":"Pure Math. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1007\/978-1-4419-6594-3_29","article-title":"Roots of AG-bands","volume":"Volume 42","author":"Gautschi","year":"2010","journal-title":"Approximation and Computation: In Honor of Gradimir V. Milovanovi\u0107, Springer Optimization and Its Applications"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Hwang, I.H., Kim, H.S., and Neggers, J. (2019). Some implicativities for groupoids and BCK-algebras. Mathematics, 7.","DOI":"10.3390\/math7100973"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Zhang, X., and Du, Y. (2022). A class of BCI-algebra and quasi-hyper BCI-algebra. Axioms, 11.","DOI":"10.3390\/axioms11020072"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Du, Y., and Zhang, X. (2022). QM-BZ-algebras and quasi-hyper BZ-algebras. Axioms, 11.","DOI":"10.3390\/axioms11030093"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Heidari, D., and Cristea, I. (2019). Breakable semihypergroups. Symmetry, 11.","DOI":"10.3390\/sym11010100"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Heidari, D., and Cristea, I. (2020). On factorizable semihypergroups. Mathematics, 8.","DOI":"10.3390\/math8071064"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/8\/378\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:00:10Z","timestamp":1760140810000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/8\/378"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,30]]},"references-count":32,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["axioms11080378"],"URL":"https:\/\/doi.org\/10.3390\/axioms11080378","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,7,30]]}}}