Geodesic
Characterizations of ordered Γ-Abel-Grassmann's groupoids
Discussiones Mathematicae. General Algebra and Applications, Tome 34 (2014) no. 1, pp. 55-73

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In this paper, we introduced the concept of ordered Γ-AG-groupoids, Γ-ideals and some classes in ordered Γ-AG-groupoids. We have shown that every Γ-ideal in an ordered Γ-AG**-groupoid S is Γ-prime if and only if it is Γ-idempotent and the set of Γ-ideals of S is Γ-totally ordered under inclusion. We have proved that the set of Γ-ideals of S form a semilattice, also we have investigated some classes of ordered Γ-AG**-groupoid and it has shown that weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular and (2,2)-regular ordered Γ-AG**-groupoids coincide. Further we have proved that every intra-regular ordered Γ-AG**-groupoid is regular but the converse is not true in general. Furthermore we have shown that non-associative regular, weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular, (2,2)-regular and strongly regular Γ-AG*-groupoids do not exist.
Keywords: ordered Γ-AG-groupoids, Γ-ideals, regular Γ-AG**-groupoids 
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Khan, Madad; Amjid, Venus; Zaman, Gul; Yaqoob, Naveed. Characterizations of ordered Γ-Abel-Grassmann's groupoids. Discussiones Mathematicae. General Algebra and Applications, Tome 34 (2014) no. 1, pp. 55-73. http://geodesic.mathdoc.fr/item/DMGAA_2014_34_1_a3/