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, we
discuss 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.

Keywords

• AG-groupoid,AG-subgroupoid,Cancellative elements,non-cancellative elements

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