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Characterizations of ordered Γ-Abel-Grassmann's groupoids

Madad Khan; Venus Amjid; Gul Zaman; Naveed Yaqoob

Discussiones Mathematicae - General Algebra and Applications (2014)

  • Volume: 34, Issue: 1, page 55-73
  • ISSN: 1509-9415

Abstract

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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.

How to cite

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Madad Khan, et al. "Characterizations of ordered Γ-Abel-Grassmann's groupoids." Discussiones Mathematicae - General Algebra and Applications 34.1 (2014): 55-73. <http://eudml.org/doc/270662>.

@article{MadadKhan2014,
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.},
author = {Madad Khan, Venus Amjid, Gul Zaman, Naveed Yaqoob},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {ordered Γ-AG-groupoids; Γ-ideals; regular Γ-AG**-groupoids},
language = {eng},
number = {1},
pages = {55-73},
title = {Characterizations of ordered Γ-Abel-Grassmann's groupoids},
url = {http://eudml.org/doc/270662},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Madad Khan
AU - Venus Amjid
AU - Gul Zaman
AU - Naveed Yaqoob
TI - Characterizations of ordered Γ-Abel-Grassmann's groupoids
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2014
VL - 34
IS - 1
SP - 55
EP - 73
AB - 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.
LA - eng
KW - ordered Γ-AG-groupoids; Γ-ideals; regular Γ-AG**-groupoids
UR - http://eudml.org/doc/270662
ER -

References

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