Korean J. Math.  Vol 23, No 3 (2015)  pp.357-370
DOI: https://doi.org/10.11568/kjm.2015.23.3.357

On $(m,n)$-ideals of an ordered Abel-Grassmann groupoid

Faisal Yousafzai, Asad Khan, Aiyared Iampan


In this paper, we introduce the concept of $(m,n)$-ideals in a non-associative ordered structure, which is called an ordered Abel-Grassmann's groupoid, by generalizing the concept of $(m,n)$-{ideals in an ordered semigroup [14]. We also study the} $(m,n)$-regular class of an ordered $\mathcal{AG}$-groupoid in terms of $(m,n)$-{ideals}.


ordered AG-groupoid, left invertive law, left identity and (m, n)-ideal.

Subject classification

20M10, 20N99.


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M. Akram, N. Yaqoob and M. Khan, On (m,n)-ideals in LA-semigroups, Ap- plied mathematical Sciences 7 (2013), 2187–2191. (Google Scholar)

P. Holgate, Groupoids satisfying a simple invertive law, The Math. Stud., 1-4, 61 (1992), 101–106. (Google Scholar)

M. A. Kazim and M. Naseeruddin, On almost semigroups, The Alig. Bull. Math. 2 (1972), 1–7. (Google Scholar)

M. Khan, F. Yousafsai and K. P. Shum, Minimal ideals of Abel Grassmann groupoids, to appear in Quasi-groups and related systems. (Google Scholar)

S. Lajos, Generalized ideals in semigroups, Acta Sci. Math. 22 (1961), 217–222. (Google Scholar)

Q. Mushtaq and S. M. Yusuf, On LA-semigroups, The Alig. Bull. Math. 8 (1978), 65–70. (Google Scholar)

Q. Mushtaq and S. M. Yusuf, On locally associative LA-semigroups, J. Nat. Sci. Math. 19 (1979), 57–62. (Google Scholar)

Q. Mushtaq and S. M. Yusuf, On LA-semigroup defined by a commutative inverse semigroups, Math. Bech. 40 (1988), 59–62. (Google Scholar)

Q. Mushtaq and M. S. Kamran, On LA-semigroups with weak associative law, Scientific Khyber 1 (1989), 69–71. (Google Scholar)

Q. Mushtaq and M. Khan, Ideals in left almost semigroups, Proceedings of 4th International Pure Mathematics Conference, (2003), 65–77. (Google Scholar)

Q. Mushtaq and M. Khan, M-systems in LA-semigroups, Southeast Asian Bull. Math. 33 (2009), 321–327. (Google Scholar)

Q. Mushtaq, M. Khan and K. P. Shum, Topological structure on LA-semigroups, Bull Malays Math. Sci. 36 (2013), 901–906. (Google Scholar)

P. V. Proti ́c and N. Stevanovi ́c, AG-test and some general properties of Abel-Grassmann’s groupoids, PU. M. A., 4, 6 (1995), 371–383. (Google Scholar)

J. Sanborisoot and T. Changphas, On Characterizations of (m,n)-regular ordered semigroups, Far East J. Math. Sci 65 (2012), 75–86. (Google Scholar)

N. Stevanovi ́c and P. V. Proti ́c, Composition of Abel-Grassmann’s 3-bands, Novi Sad, J. Math., 2, 34 (2004), 175–182. (Google Scholar)

X. Y. Xie and J. Tang, Fuzzy radicals and prime fuzzy ideals of ordered semi-groups, Inform. Sci. 178 (2008), 4357–4374. (Google Scholar)

F. Yousafzai, N. Yaqoob and A. Ghareeb, Left regular AG-groupoids in terms of fuzzy interior ideals, Afrika Mathematika 24 (2013), 577–587. (Google Scholar)

F. Yousafzai, A. Khan and B. Davvaz, On fully regular AG-groupoids, Afrika Mathematika 25 (2014), 449–459. (Google Scholar)

F. Yousafzai, A. Khan, V. Amjad and A. Zeb, On fuzzy fully regular ordered AG-groupoids, Journal of Intelligent & Fuzzy Systems 26 (2014), 2973–2982. (Google Scholar)


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