Cancellative Elements in Finite AG-groupoids

Abstract

An Abel-Grassmann's groupoid (brie y AG-groupoid) is a groupoid S satisfying the left invertive law: (xy)z = (zy)x for all x, y, z \in S. In the present paper, wediscuss the left and right cancellative property of elements of the nite AG-groupoid S. For an AG-groupoid with left identity it is known that every left cancellative ele-ment is right cancellative. We prove a problem (for nite AG-groupoids) that every left cancellative element of an AG-groupoid (without left identity) is right cancella-tive. Moreover, we generalize various results of nite AG-groupoids by removing the condition of existence of left identity.