A characterization of ordered Γ-semigroups by ordered (m,n)-Γ-ideals
DOI:
https://doi.org/10.5269/bspm.41186Abstract
In this paper, we study (m,n)-regular ordered Γ-semigroups through ordered (m,n)-Γ-ideals. It is shown that if (S,Γ,·,≤) is an ordered Γ-semigroup; m,n are non-negative integers and A(m,n) is the set of all ordered (m,n)-Γ-ideals of S. Then, S is (m,n)-regularâ‡â‡’ ∀A ∈ A(m,n), A = (AmΓSΓAn]. It is also proved that if (S,Γ,·,≤) is an ordered Γ-semigroup and m,n are nonnegative integers and R(m,0) and L(0,n) is the set of all (m,0)-Γideals and (0,n)-Γ-ideals of S, respectively. Then, S is (m,n)-regular ordered Γ semigroup â‡â‡’∀R ∈R(m,0)∀L ∈L(0,n),R∩L = (RmΓL∩RΓLn].
References
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19. Yaqoob, N., Aslam, M., Davvaz, B. and Saeid, A. B. , On rough (m,n) bi-ô€€€-hyperideals in ô€€€-semihypergroups, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 75(1) (2013) 119-128.
20. Yaqoob, N. and Chinram, R., 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.
21. Yousafza, I. F., Khan, W. Guo and Khan, M., On (m,n)-ideals of left almost semigroups, Journal of Semigroup Theory and Applications, (2014) 1-20.
22. Yousafzai, F., Khan, A. and Iampan, A., On (m,n)-ideals of an ordered Abel-Grassmann groupoid, The Korean Journal of Mathematics, 23(3) (2015) 357-70.
2. Abbasi, M. Y. and Basar Abul, Some properties of ordered 0-minimal (0; 2)-bi-ô€€€-ideals in po-ô€€€-semigroups, Hacettepe Journal of Mathematics and Statistics, 44(2) (2015),247 -254.
3. Abbasi, M. Y. and Basar Abul, A note on ordered quasi-ô€€€-ideals in intra-regular po-ô€€€- semigroups, International Journal of Pure and Applied Mathematics, 2(98) (2015),221-230.
4. Akram, M. , N. Yaqoob, N. and Khan, M., On (m,n)-ideals in LA-semigroups, Applied Mathematical Sciences, 7(44) (2013) 2187-2191.
5. Amjad, V., Hila, K. and Yousafzai, F., Generalized hyperideals in locally associative left almost semihypergroups, New York Journal of Mathematics, 20 (2014) 1063–1076.
6. Aslam, M., Shabir, M. Yaqoob, N. and Shabir, A. , On rough (m,n)-bi-ideals and generalized rough (m,n)-bi-ideals in semigroups, Annals of Fuzzy Mathematics and Informatics, 2(2) (2011) 141-150.
7. Bussaban, L. and Changphas, T., On (m,n)-ideals and (m,n)-regular ordered semigroups, Songklanakarin J. Sci. Technol. 38 (2) (2016) 199-206.
8. Basar, A. and Abbasi, M. Y., On generalized bi-ô€€€-ideals in ô€€€-semigroups, Quasigroups and Related Systems, 23(2) (2015).
9. Changphas, T., Generalized ideals in ordered semigroups, Far East Journal of Mathematical Sciences, 81(2) (2013) 147-154.
10. Gaketem, ON (m, n)-Reguler in LA-semigroups, Far East Journal of Mathematical Sciences, 96(4) (2015) 409.
11. Iampan, A. and Siripitukdet, M., On minimal and maximal ordered left ideals in po-ô€€€-semigroups, Thai Journal of Mathematics, 2 (2004), 275-282.
12. Khan, W., Yousafzai, F. and Khan, M., On (m,n)-ideals of left almost semigroups, European Journal of Pure and Applied Mathematics, 9(3) (2016) 277-291.
13. Kehayopulu, N., On prime, weakly prime ideals in po-ô€€€-semigroups, Lobachevskii Journal of Mathematics, 30(4)(2009), 257-262.
14. Krgovic, D. N., On (m, n)-regular semigroups, Publications De L’ Institut Mathematique, 18(32) (1975), 107-110.
15. Kwon, Y. I. and Lee, S. K., Some special elements in ordered ô€€€-semigroups, Kyungpook Math. J., 35(1996), 679-685.
16. Lajos, S., Generalized ideals in semigroups, Acta Sci. Math., 22 (1961), 217–222.
17. Sen, M. K., On ô€€€-semigroups, Algebra and its applications, Int. Symp., New Delhi, (1981), Lecture Notes in Pure and Applied Mathematics 91, Decker, New York,(1984), 301–308.
18. Yaqoob, N. and Aslam, M., Prime (m,n) bi-ô€€€-hyperideals in ô€€€-semihypergroups, Applied Mathematics and Information Sciences, 8(5) (2014) 2243-2249.
19. Yaqoob, N., Aslam, M., Davvaz, B. and Saeid, A. B. , On rough (m,n) bi-ô€€€-hyperideals in ô€€€-semihypergroups, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 75(1) (2013) 119-128.
20. Yaqoob, N. and Chinram, R., 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.
21. Yousafza, I. F., Khan, W. Guo and Khan, M., On (m,n)-ideals of left almost semigroups, Journal of Semigroup Theory and Applications, (2014) 1-20.
22. Yousafzai, F., Khan, A. and Iampan, A., On (m,n)-ideals of an ordered Abel-Grassmann groupoid, The Korean Journal of Mathematics, 23(3) (2015) 357-70.
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2020-10-10
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