In contemporary mathematics, parameterization tool like soft set theory precisely tackle complex problems of economics and engineering. In this paper, we demonstrate a novel approach of soft set theory i.e., intersectional soft (int-soft) sets of an ordered $ \Gamma $ -semigroup $ S $ and develop int-soft left (resp. right) $ \Gamma $-ideals of $ S. $ Various classes like $ \Gamma $-regular, left $ \Gamma $-simple, right $ \Gamma $-simple and some semilattices of an ordered $ \Gamma $-semigroup $ S $ are characterize through int-soft left (resp. right) $ \Gamma $-ideals of $ S. $ Particularly, a $ \Gamma $-regular ordered $ \Gamma $-semigroup $ S $ is a left $ \Gamma $-simple if and only if every int-soft left $ \Gamma $-ideal $ f_{A} $ of $ S $ is a constant function. Also, $ S $ is a semilattice of left (resp. right) $ \Gamma $-simple $ \Gamma $-semigroup if and only if for every int-soft left (resp. right) $ \Gamma $-ideal $ f_{A} $ of $ S, $ $ f_{A}\left(a\right) = f_{A}\left(a\alpha a\right) $ and $ f_{A}\left(a\alpha b\right) = f_{A}\left(b\alpha a\right) $ for all $ a, b\in S $ and $ \alpha \in \Gamma $ hold.
Citation: Faiz Muhammad Khan, Tian-Chuan Sun, Asghar Khan, Muhammad Junaid, Anwarud Din. Intersectional soft gamma ideals of ordered gamma semigroups[J]. AIMS Mathematics, 2021, 6(7): 7367-7385. doi: 10.3934/math.2021432
In contemporary mathematics, parameterization tool like soft set theory precisely tackle complex problems of economics and engineering. In this paper, we demonstrate a novel approach of soft set theory i.e., intersectional soft (int-soft) sets of an ordered $ \Gamma $ -semigroup $ S $ and develop int-soft left (resp. right) $ \Gamma $-ideals of $ S. $ Various classes like $ \Gamma $-regular, left $ \Gamma $-simple, right $ \Gamma $-simple and some semilattices of an ordered $ \Gamma $-semigroup $ S $ are characterize through int-soft left (resp. right) $ \Gamma $-ideals of $ S. $ Particularly, a $ \Gamma $-regular ordered $ \Gamma $-semigroup $ S $ is a left $ \Gamma $-simple if and only if every int-soft left $ \Gamma $-ideal $ f_{A} $ of $ S $ is a constant function. Also, $ S $ is a semilattice of left (resp. right) $ \Gamma $-simple $ \Gamma $-semigroup if and only if for every int-soft left (resp. right) $ \Gamma $-ideal $ f_{A} $ of $ S, $ $ f_{A}\left(a\right) = f_{A}\left(a\alpha a\right) $ and $ f_{A}\left(a\alpha b\right) = f_{A}\left(b\alpha a\right) $ for all $ a, b\in S $ and $ \alpha \in \Gamma $ hold.
| [1] | D. Molodtsov, Soft set theory-first results, Comput. Math. Appl., 37 (1999), 19–31. |
| [2] |
P. K. Maji, A. R. Roy, R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl., 44 (2002), 1077–1083. doi: 10.1016/S0898-1221(02)00216-X
|
| [3] | D. Molodtsov, V. Yu. Leonov, D. V. Kovkov, Soft set technique and its applications, Nechetkie Sistemy i Myagkie Vychisleniya, 1 (2006), 8–39. |
| [4] |
A. R. Roy, P. K. Maji, A soft set theoretic approach to decision making problems, J. Comput. Appl. Math., 203 (2007), 412–418. doi: 10.1016/j.cam.2006.04.008
|
| [5] | Y. B. Jun, C. H. Park, Applications of soft sets in ideal theory, Inform. Sciences, 178 (2008), 2466–2475. |
| [6] |
Y. Zou, Z. Xiao, Data analysis approaches of soft sets under incomplete information, Knowl.-Based Syst., 21 (2008), 941–945. doi: 10.1016/j.knosys.2008.04.004
|
| [7] |
F. Feng, Y. B. Jun, X. Liu, L. Li, An adjustable approach to fuzzy soft set based decision making, J. Comput. Appl. Math., 234 (2010), 10–20. doi: 10.1016/j.cam.2009.11.055
|
| [8] |
N. Cagman, S. Enginoglu, Soft matrix theory and its decision making, Comput. Math. Appl., 59 (2010), 3308–3314. doi: 10.1016/j.camwa.2010.03.015
|
| [9] |
N. Cagman, S. Enginoglu, Soft set theory and uni-int decision making, Eur. J. Oper. Res., 207 (2010), 848–855. doi: 10.1016/j.ejor.2010.05.004
|
| [10] | N. Cagman, S. Enginoglu, FP-soft set theory and its applications, Ann. Fuzzy Math. Inform., 2 (2011), 219–226. |
| [11] | F. Feng, Soft rough sets applied to multicriteria group decision making, Ann. Fuzzy Math. Inform., 2 (2011), 69–80. |
| [12] | P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Comput. Math. Appl., 45 (2003), 555–562. |
| [13] |
M. I. Ali, F. Feng, W. K. Min, M. Shabir, On some new operations in soft set theory, Comput. Math. Appl., 57 (2009), 1547–1553. doi: 10.1016/j.camwa.2008.11.009
|
| [14] |
M. I. Ali, M. Shabir, M. Naz, Algebraic structure of soft sets associated with new operations, Comput. Math. Appl., 61 (2011), 2647–2654. doi: 10.1016/j.camwa.2011.03.011
|
| [15] | L. Fuchs, Partially ordered algebraic systems, Oxford, London, NewYork, Paris: Dover Publications, 1963. |
| [16] | N. Kehayopulu, On filters generalized in poe-semigroups, Mathematica Japonica, 35 (1990), 789–796. |
| [17] | N. Kehayopulu, Remarks on ordered semigroups, Mathematica Japonica, 35 (1990), 1061–1063. |
| [18] |
N. Kehayopulu, M. Tsingelis, Fuzzy sets in ordered groupoid, Semigroup Forum, 65 (2002), 128–132. doi: 10.1007/s002330010079
|
| [19] | M. K. Sen, N. K. Saha, On $\Gamma $-semigroups-Ⅰ, Bulletin of the Calcutta Mathematical Society, 78 (1986), 180–186. |
| [20] | N. K. Saha, On $\Gamma $-semigroup-Ⅱ, Bulletin of the Calcutta Mathematical Society, 79 (1987), 331–335. |
| [21] | N. K. Saha, On $\Gamma $-semigroup-Ⅲ, Bulletin of the Calcutta Mathematical Society, 80 (1988), 1–12. |
| [22] | Y. I. Kwon, S. K. Lee, On weakly prime ideals of ordered -semigroups, Communications of the Korean Mathematical Society, 13 (1998), 251–256. |
| [23] | R. Chinram, C. Jirojkul, On bi $\Gamma $-ideals in $\Gamma $ -semigroups, Songklanakarin Journal of Science and Technology, 29 (2007), 231–234. |
| [24] | T. K. Dutta, N. C. Adhikari, On prime radical of $\Gamma $-semigroups, Bulletin of the Calcutta Mathematical Society, 86 (1994), 437–444. |
| [25] | M. Shabir, S. Ali, Prime bi-ideals of $\Gamma $-semigroups, JARPM, 4 (2012), 47–58. |
| [26] | D. R. Prince William, K. B. Latta, E. Chandrasekeran, Fuzzy bi-$\Gamma $-ideals in semigroups, Hacettepe Journal of Mathematics and Statistics, 38 (2009), 1–15. |
| [27] |
M. Uckan, A. O. Mehmet, Y. B. Jun, Intuitionistic fuzzy sets in -semigroup, B. Korean Math. Soc., 44 (2007), 359–367. doi: 10.4134/BKMS.2007.44.2.359
|
| [28] | Y. B. Jun, M. Sapanci, M. A. Ozturk, Fuzzy ideals in gamma near-rings, Turk. J. Math., 22 (1998), 449–459. |
| [29] | Y. B. Jun, C. Y. Lee, Fuzzy-rings, EAMJ, 8 (1992), 163–170. |
| [30] | A. Iampan, Characterizing ordered bi-Ideals in ordered $\Gamma $ -semigroups, IJMSI, 4 (2009), 17–25. |
| [31] | Y. B. Jun, S. Z. Song, Int-soft (generalized) bi-ideals of semigroups, Sci. World J., 2015 (2015), 1–6. |
| [32] | S. Z. Song, H. S. Kim, Y. B. Jun, Ideal theory in semigroups based on inter sectional soft sets, Sci. World J., 2014 (2014), 1–7. |
| [33] |
A. Khan, M. Sarwar, Uni-soft bi-ideals and uni-soft interior ideals of AG-groupoids, Math. Sci. Lett., 5 (2016), 271–277. doi: 10.18576/msl/050308
|
| [34] | M. Sarwar, A. Khan, On uni-soft (quasi) ideals of AG-groupoids, Appl. Math. Sci., 8 (2014), 589–600. |
| [35] | Y. B. Jun, S. Z. Song, G. Muhiuddin, Concave soft sets, critical soft points, and union-soft ideals of ordered semigroups, Sci. World J., 2014 (2014), 1–11. |
| [36] | A. Khan, R. Khan, Y. B. Jun, Uni-soft structure applied to ordered semigroups, Soft Comput., 21 (2015), 1021–1030. |
| [37] | E. H. Hamouda, Soft ideals in ordered semigroups, Rev. Union Mat. Argent., 58 (2017), 85–94. |
| [38] | G. Muhiuddin, A. Mahboob, Int-soft ideals over the soft sets in ordered semigroups, AIMS Mathematics, 5 (2020): 2412–2423. |
| [39] | J. Ghosh, D. Mandal, T. K. Samanta, Soft prime and semiprime int-ideals of a ring, AIMS Mathematics, 5 (2019), 732–745. |
| [40] | W. A. Khan, A. Rehman, A. Taouti, Soft near-semirings, AIMS Mathematics, 5 (2020), 6464–6478. |
| [41] |
A. S. Sezer, N. Çağman, A. O. Atagün, M. I. Ali, E. Türkmen, Soft intersection semigroups, ideals and bi-Ideals: a new application on semigroup theory, Filomat, 29 (2015), 917–946. doi: 10.2298/FIL1505917S
|
| [42] |
X. Ma, J. Zhan, B. Davvaz, Applications of soft intersection sets to hemirings via SI-h-bi-ideals and SI-h-quasi-ideals, Filomat, 30 (2016), 2295–2313. doi: 10.2298/FIL1608295M
|
| [43] |
F. Ctak, N. CaGman, Soft k-int-ideals of semirings and its algebraic structures, Ann. Fuzzy Math. Inform., 13 (2017), 531–538. doi: 10.30948/afmi.2017.13.4.531
|
| [44] |
W. A. Khan, B. Davvaz, A. Muhammad, (M, N)-soft intersection nearsemirings and (M, N)-$\alpha $-inclusion along with its algebraic applications, Lobachevskii Journal of Mathematics, 40 (2019), 67–78. doi: 10.1134/S1995080219010098
|
| [45] | A. F. Talee, M. Y. Abbasi, K. Hila, S. A. Khan, A new approach towards int-soft hyperideals in ordered ternary semihypergroups, Journal of Discrete Mathematical Sciences and Cryptography, 2020, DOI: 10.1080/09720529.2020.1747674. |
| [46] | U. Acar, F. Koyuncu, B. Tanay, Soft sets and soft rings, Comput. Math. Appl., 59 (2010), 3458–3463. |
| [47] |
A. O. Atagün, A. Sezgin Soft substructures of rings, fields and modules, Comput. Math. Appl., 61 (2011), 592–601. doi: 10.1016/j.camwa.2010.12.005
|
| [48] | H. Aktaş, N. Çağman, Soft sets and soft groups, Inform. Sciences, 177 (2007), 2726–2735. |
| [49] | F. Feng, Y. B. Jun, X. Zhao, Soft semirings, Comput. Math. Appl., 56 (2008), 2621–2628. |
| [50] | Y. B. Jun, K. J. Lee, A. Khan, Soft ordered semigroups, Math. Logic Quart., 56 (2010), 42–50. |
| [51] | Y. B. Jun C. H. Park, N. O. Alshehri, Hypervector spaces based on intersectional soft sets, Abstr. Appl. Anal., 2014 (2014), 1–6. |
| [52] | A. Sezgin, A. O. Atagun, On operations of soft sets, Comput. Math. Appl., 61 (2011), 1457–1467. |
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Faiz Muhammad Khan, Tian-Chuan Sun, Asghar Khan, Muhammad Junaid, Anwarud Din. Intersectional soft gamma ideals of ordered gamma semigroups[J]. AIMS Mathematics, 2021, 6(7): 7367-7385. doi: 10.3934/math.2021432
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