Characterizations of ordered Γ-Abel-Grassmann's groupoids
Journal Title: Discussiones Mathematicae - General Algebra and Applications - Year 2014, Vol 34, Issue 1
Abstract
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. 2010 Mathematics Subject Classification: 20M10, 20N99.
Authors and Affiliations
Madad Khan, Venus Amjid, Gul Zaman, Naveed Yaqoob
Keywords
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