A Note on $(m, n)$-$\Gamma$-Ideals of Ordered $LA$-$\Gamma$-Semigroups
A Note on $(m, n)$-$\Gamma$-Ideals of Ordered $LA$-$\Gamma$-Semigroups
In this paper, we investigate the notion of $(m, n)$-ideals in a non-associative algebraic structure, which we call an ordered $LA$-$\Gamma$-semigroup. We prove that if $(S, \Gamma, \cdot, \leq)$ is a unitary ordered $LA$-$\Gamma$-semigroup with zero and $S$ has the condition that it contains no non-zero nilpotent $(m, n)$-ideals and if $R(L)$ is a 0-minimal right (left) ideal of $S$, then either $(R\Gamma L]=\{0\}$ or $(R\Gamma L]$ is a 0-minimal $(m, n)$-ideal of $S$. Also, we prove that if $(S, \Gamma, \cdot, \leq)$ is a unitary ordered $LA$-$\Gamma$-semigroup; $A$ is an $(m, n)$-ideal of $S$ and $B$ is an $(m, n)$-ideal of $A$ such that $B$ is idempotent, then $B$ is an $(m, n)$-ideal of $S$.
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- [1] Abul Basar and M. Y. Abbasi, Some Properties of Q-Fuzzy Ideals in po-G-Semigroups, Palestine Journal of Mathematics, 7(2)(2018), 505–511.
- [2] Abul Basar, A note on intra-regular ordered G-semigroups with invo-lution, Eurasian Bulletin of Mathematics, 1(2) (2018), 78-84.
- [3] Abul Basar and M. Y. Abbasi, On covered G-ideals in G-semigroups, Global Journal of Pure and Applied Mathematics, 13(5)(2017), 1465–1472.
- [4] Abul Basar and M. Y. Abbasi, On generalized bi-G-ideals in G-semigroups, Quasigroups And Related Systems, 23(2015), 181–186.
- [5] F. Yousafzai, A. Khan and B. Davvaz, On fully regular AG-groupoids, Afrika Mathematica, 25(2014), 449-459.
- [6] F. Yousafzai, A. Khan, and A. Iampan, On (m;n)-ideals of an ordered Abel-Grassmann Groupoid, Korean J. Math., 3(23)(2015), 357-370.
- [7] M. Akram, N. Yaqoob and M. Khan, On (m;n)-ideals in LA-semigroups, Applied mathematical Sciences, 7(2013), 2187 - 2191.
- [8] M. A. Kazim and M. Naseeruddin, On almost semigroups, The Alig. Bull. Math., 2 (1972), 1-7.
- [9] M. Y. Abbasi and Abul Basar, A note on ordered bi-G-ideals in intra-regular ordered G-semigroups, Afrika Mathematica, (27)(7-8) (2016), 1403–1407.
- [10] M. Y. Abbasi and Abul Basar, On Generalizations of ideals in LA-G-semihypergroups, Southeast Asian Bulletin of Mathematics, (39) (2015), 1-12.
- [11] M. Y. Abbasi and A. Basar, Some properties of ordered 0-minimal (0, 2)-bi-G-ideals in po-G-semigroups, Hacettepe Journal of Mathematics and
Statistics, 2(44)(2015), 247–254.
- [12] M. Y. Abbasi and Abul Basar, Weakly prime ideals in involution po-G-semigroups, Kyungpook Mathematical Journal, 54(2014), 629-638.
- [13] N. Yaqoob and M. Aslam, Prime (m,n)bi-G-hyperideals in G-semihypergroups, Applied Mathematics and Information Sciences, 8(5) (2014), 2243-2249.
- [14] N. Yaqoob, M. Aslam, B. Davvaz and A. B. Saeid, On rough (m,n)bi-G-hyperideals in G-semigroups, UPB Scientific Bulletin, Series A: Applied
Mathematics and Physics, 1 (75) (2013), 119-128.
- [15] N. Yaqoob and R. Chinram, On prime (m,n)bi-ideals and rough prime (m,n)bi-ideals in semigroups, Far East Journal of Mathematical Sciences, 62(2)
(2012), 145-159.
- [16] Q. Mushtaq and S. M. Yusuf, On locally associative LA-semigroups,J. Nat. Sci. Math.,(19)(1979), 57-62.
- [17] Satyanarayana Bhavanari, M. Y. Abbasi, Abul Basar and Syam Prasad Kuncham, Some Results on Abstract Affine Gamma-Near-Rings, International
Journal of Pure and Applied Mathematical Sciences, 7(1) (2014), 43–49.
- [18] T. Shah and I. Rehman, On G-ideals and bi-G-ideals in G-AG-groupoid, International Journal of Algebra, 6(4)(2010), 267-276.
- [19] V. Amjad, K. Hila and F. Yousafzai, Generalized hyperideals in locally associative left almost semihypergroups, New York Journal of Mathematics,
(20)(2014), 1063-1076.
- [20] W. Khan, F. Yousafzai and M. Khan, On (m,n)-ideals of left almost semigroups, European Journal of Pure and Applied Mathematics, 9(3)(2016),
277-291.