Transposition Regular AG-Groupoids and Their Decomposition Theorems

Yudan Du, Xiaohong Zhang and Xiaogang An
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Yudan Du: School of Mathematics & Data Science, Shaanxi University of Science & Technology, Xi’an 710021, China
Xiaohong Zhang: School of Mathematics & Data Science, Shaanxi University of Science & Technology, Xi’an 710021, China
Xiaogang An: School of Mathematics & Data Science, Shaanxi University of Science & Technology, Xi’an 710021, China

Mathematics, 2022, vol. 10, issue 9, 1-20

Abstract: In this paper, we introduce transposition regularity into AG-groupoids, and a variety of transposition regular AG-groupoids (L1/R1/LR, L2/R2/L3/R3-groupoids) are obtained. Their properties and structures are discussed by their decomposition theorems: (1) L1/R1-transposition regular AG-groupoids are equivalent to each other, and they can be decomposed into the union of disjoint Abelian subgroups; (2) L1/R1-transposition regular AG-groupoids are LR-transposition regular AG-groupoids, and an example is given to illustrate that not every LR-transposition regular AG-groupoid is an L1/R1-transposition regular AG-groupoid; (3) an AG-groupoid is an L1/R1-transposition regular AG-groupoid if it is an LR-transposition regular AG-groupoid satisfying a certain condition; (4) strong L2/R3-transposition regular AG-groupoids are equivalent to each other, and they are union of disjoint Abelian subgroups; (5) strong L3/R2-transposition regular AG-groupoids are equivalent to each other and they can be decomposed into union of disjoint AG subgroups. Their relations are discussed. Finally, we introduce various transposition regular AG-groupoid semigroups and discuss the relationships among them and the commutative Clifford semigroup as well as the Abelian group.

Keywords: semigroup; transposition regular AG-groupoid; decomposition theorem; Abelian group (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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