International Journal of Mathematics and Mathematical Sciences
Volume 27 (2001), Issue 6, Pages 387-389

Finite AG-groupoid with left identity and left zero

Qaiser Mushtaq and M. S. Kamran

Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

Received 3 October 2000

Copyright © 2001 Qaiser Mushtaq and M. S. Kamran. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A groupoid G whose elements satisfy the left invertive law: (ab)c=(cb)a is known as Abel-Grassman's groupoid (AG-groupoid). It is a nonassociative algebraic structure midway between a groupoid and a commutative semigroup. In this note, we show that if G is a finite AG-groupoid with a left zero then, under certain conditions, G without the left zero element is a commutative group.