On Introduction of New Classes of AG-groupoids

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On Introduction of New Classes of AG-groupoids

Author Affiliations

¹Department of Mathematics, Quaid-e-Azam University, Islamabad, PAKISTAN
² Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, PAKISTAN

Res. J. Recent Sci., Volume 2, Issue (1), Pages 67-70, January,2 (2013)

Abstract

We discover eight new subclasses of AG-groupoids namely; Type-1, Type-2, Left Type-3, Right Type-3, Type-3, Backward Type-4, Forward Type-4, and Type-4. We provide examples of each type to prove their existence. We also give their enumer-ation up to order 6 and prove some of their basic properties and relations with each other and with other known classes.

References

Kazim M.A. and Naseerudin M., On almost semigroups, Portugaliae Mathematica,36(1), (1977)
Cho J.R., Pusan Jezek J. and Kepka T., Praha, Paramedial Groupoids, Czechoslovak Mathematical Journal, 49(124), (1996) Praha
Stevanovic N. and Protic P.V., Abel-grassmann’s bands, Quasigroups and Related Systems, 11(1) 95–101 2004.
Stevanovic N. and Protic P.V., Composition of Abel-Grassmann’s 3-bands, Novi Sad J. Math., 34(2), 175182 (2004)
Shah M., A theoretical and computational investigations of AG-groups, PhD thesis, Quaid-i-Azam University Islamabad, (2012)
Shah M. and Ali A., Some structural properties of AG-group, International Mathematical Forum 6, 34, 1661–1667 (2011)
Naseeruddin M., Some studies on almost semigroups and flocks, Ph.D Thesis, The Aligarh Muslim University, India, (1970)
Mushtaq Q. and Yusuf S.M., On Locally Associative LA-semigroup, J. Nat. Sci. Math. Vol. XIX, No.1, 57–62 (1979)
Shah M., Shah T. and A. Ali, On the cancellativity of AG-groupoids, International Mathematical Forum 6, 44, 2187–2194 (2011)
Howie J.M., Fundamentals of Semigroup Theory, Clarendon Press, Oxford, (2003)
Shah M., Ahmad I. and A. Ali, Discovery of new classes of AG-groupoids, Res. J. Recent Sci., 1(11), 47-49 (2012)
Mushtaq Q. and Yusuf S.M., On LA-semigroups, Alig. Bull. Math., 8(1), 65-70 (1978)
Mushtaq Q. and Kamran M.S., On left almost groups, Proc. Pak. Acad. of Sciences, 33, 1–2 (1996)

[ref1] Kazim M.A. and Naseerudin M., On almost semigroups, Portugaliae Mathematica,36(1), (1977)

[ref2] Cho J.R., Pusan Jezek J. and Kepka T., Praha, Paramedial Groupoids, Czechoslovak Mathematical Journal, 49(124), (1996) Praha

[ref3] Stevanovic N. and Protic P.V., Abel-grassmann’s bands, Quasigroups and Related Systems, 11(1) 95–101 2004.

[ref4] Stevanovic N. and Protic P.V., Composition of Abel-Grassmann’s 3-bands, Novi Sad J. Math., 34(2), 175182 (2004)

[ref5] Shah M., A theoretical and computational investigations of AG-groups, PhD thesis, Quaid-i-Azam University Islamabad, (2012)

[ref6] Shah M. and Ali A., Some structural properties of AG-group, International Mathematical Forum 6, 34, 1661–1667 (2011)

[ref7] Naseeruddin M., Some studies on almost semigroups and flocks, Ph.D Thesis, The Aligarh Muslim University, India, (1970)

[ref8] Mushtaq Q. and Yusuf S.M., On Locally Associative LA-semigroup, J. Nat. Sci. Math. Vol. XIX, No.1, 57–62 (1979)

[ref9] Shah M., Shah T. and A. Ali, On the cancellativity of AG-groupoids, International Mathematical Forum 6, 44, 2187–2194 (2011)

[ref10] Howie J.M., Fundamentals of Semigroup Theory, Clarendon Press, Oxford, (2003)

[ref11] Shah M., Ahmad I. and A. Ali, Discovery of new classes of AG-groupoids, Res. J. Recent Sci., 1(11), 47-49 (2012)

[ref12] Mushtaq Q. and Yusuf S.M., On LA-semigroups, Alig. Bull. Math., 8(1), 65-70 (1978)

[ref13] Mushtaq Q. and Kamran M.S., On left almost groups, Proc. Pak. Acad. of Sciences, 33, 1–2 (1996)