A new method to evaluate regular ternary semigroups in multi-polar fuzzy environment

Shahida Bashir; Ahmad N. Al-Kenani; Maria Arif; Rabia Mazhar; Shahida Bashir; Ahmad N. Al-Kenani; Maria Arif; Rabia Mazhar

doi:10.3934/math.2022680

AIMS Mathematics

2022, Volume 7, Issue 7: 12241-12263. doi: 10.3934/math.2022680

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A new method to evaluate regular ternary semigroups in multi-polar fuzzy environment

1.
Department of mathematics, University of Gujrat, Gujrat 50700, Pakistan
2.
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80219, Jeddah 21589, Saudi Arabia

Received: 28 October 2021 Revised: 09 March 2022 Accepted: 14 March 2022 Published: 24 April 2022
MSC : 03E72, 18B40, 20N10

Theory of $m$-polar fuzzy set deals with multi-polar information. It is used when data comes from $m$ factors $\left({m \ge 2} \right)$. The primary objective of this work is to explore a generalized form of $m$-polar fuzzy subsemigroups, which is $m$-polar fuzzy ternary subsemigroups. There are many algebraic structures which are not closed under binary multiplication that is a reason to study ternary operation of multiplication such as the set of negative integer is closed under the operation of ternary multiplication but not closed for the binary multiplication. This paper, presents several significant results related to the notions of $m$-polar fuzzy ternary subsemigroups, $m$-polar fuzzy ideals, $m$-polar fuzzy generalized bi-ideals, $m$-polar fuzzy bi-ideals, $m$-polar fuzzy quasi-ideals and $m$-polar fuzzy interior ideals in ternary semigroups. Also, it is proved that every $m$- polar fuzzy bi-ideal of ternary semigroup is an $m$-polar fuzzy generalized bi-ideal of ternary semigroup but converse is not true in general. Moreover, this paper characterizes regular and intra-regular ternary semigroups by the properties of $m$-polar fuzzy ideals, $m$-polar fuzzy bi-ideals.
- ternary semigroups,
- $m$-polar fuzzy set,
- $m$-polar fuzzy ideals,
- regular and intra-regular ternary semigroups
Citation: Shahida Bashir, Ahmad N. Al-Kenani, Maria Arif, Rabia Mazhar. A new method to evaluate regular ternary semigroups in multi-polar fuzzy environment[J]. AIMS Mathematics, 2022, 7(7): 12241-12263. doi: 10.3934/math.2022680

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Abstract

Theory of $m$-polar fuzzy set deals with multi-polar information. It is used when data comes from $m$ factors $\left({m \ge 2} \right)$. The primary objective of this work is to explore a generalized form of $m$-polar fuzzy subsemigroups, which is $m$-polar fuzzy ternary subsemigroups. There are many algebraic structures which are not closed under binary multiplication that is a reason to study ternary operation of multiplication such as the set of negative integer is closed under the operation of ternary multiplication but not closed for the binary multiplication. This paper, presents several significant results related to the notions of $m$-polar fuzzy ternary subsemigroups, $m$-polar fuzzy ideals, $m$-polar fuzzy generalized bi-ideals, $m$-polar fuzzy bi-ideals, $m$-polar fuzzy quasi-ideals and $m$-polar fuzzy interior ideals in ternary semigroups. Also, it is proved that every $m$- polar fuzzy bi-ideal of ternary semigroup is an $m$-polar fuzzy generalized bi-ideal of ternary semigroup but converse is not true in general. Moreover, this paper characterizes regular and intra-regular ternary semigroups by the properties of $m$-polar fuzzy ideals, $m$-polar fuzzy bi-ideals.

References

[1]	S. S. Raju, K. V. R. Srinivas, Ternary semigroup of a lattice, Int. J. Pure Appl. Math., 119 (2018), 115–121.
[2]	D. H. Lehmer, A ternary analogue of abelian groups, Amer. J. Math., 54 (1932), 329–338. https://doi.org/10.2307/2370997 doi: 10.2307/2370997
[3]	A. Chronowski, Ternary semigroups and ternary algebras, In: Functional equations in mathematical analysis, New York: Springer, 2011,371–416. https://doi.org/10.1007/978-1-4614-0055-4_28
[4]	M. L. Santiago, S. S. Bala, Ternary semigroups, Semigroup Forum, 81 (2010), 380–388. https://doi.org/10.1007/s00233-010-9254-x doi: 10.1007/s00233-010-9254-x
[5]	J. Los, On the extending of models (Ⅰ), Fund. Math., 42 (1955), 38–54. https://doi.org/10.4064/fm-42-1-38-54 doi: 10.4064/fm-42-1-38-54
[6]	F. M. Sioson, Ideal theory in ternary semigroups, Math. Japon, 10 (1965), 63.
[7]	S. Bashir, M. Shabir, Prime left ideals in ternary semigroups, J. Adv. Res. Pure Math., 1 (2009), 1–13.
[8]	S. Bashir, M. Shabir, N. Rehman, Pure fuzzy ideals in ternary semigroups, Int. J. Algebra Stat., 1 (2012), 1–7.
[9]	S. Bashir, M. Shabir, Pure ideals in ternary semigroups, Quasigroups Related Syst., 17 (2009), 111–122.
[10]	M. Shabir, S. Bashir, Prime ideals in ternary semigroups, Asian-Eur. J. Math, 2 (2009), 141–154. https://doi.org/10.1142/S1793557109000121 doi: 10.1142/S1793557109000121
[11]	S. Bashir, X. K. Du, On weakly regular fuzzy ordered ternary semigroups, Appl. Math. Inform. Sci., 10 (2016), 2247–2254. https://doi.org/10.18576/amis/100627 doi: 10.18576/amis/100627
[12]	S. Bashir, X. K. Du, Intra-regular and weakly regular ordered ternary semigroups, Ann. Fuzzy Math. Inform., 13 (2017), 539–551.
[13]	S. Bashir, M. Fatima, M. Shabir, Regular ordered ternary semigroups in terms of bipolar fuzzy ideals, Mathematics, 7 (2019), 233. https://doi.org/10.3390/math7030233 doi: 10.3390/math7030233
[14]	S. Bashir, R. Mazhar, H. Abbas, M. Shabir, Regular ternary semirings in terms of bipolar fuzzy ideals, Comp. Appl. Math., 39 (2020), 319. https://doi.org/10.1007/s40314-020-01319-z doi: 10.1007/s40314-020-01319-z
[15]	S. Bashir, H. Abbas, R. Mazhar, M. Shabir, Rough fuzzy ternary subsemigroups based on fuzzy ideals with three-dimensional congruence relation, Comp. Appl. Math., 39 (2020), 90. https://doi.org/10.1007/s40314-020-1079-y doi: 10.1007/s40314-020-1079-y
[16]	L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
[17]	M. Shabir, S. Liaquat, S. Bashir, Regular and intra-regular semirings in terms of bipolar fuzzy ideals, Comp. Appl. Math., 38 (2019), 197. https://doi.org/10.1007/s40314-019-0974-6 doi: 10.1007/s40314-019-0974-6
[18]	M. Shabir, T. Abbas, S. Bashir, R. Mazhar, Bipolar fuzzy hyperideals in regular and intra-regular semihypergroups, Comp. Appl. Math., 40 (2021), 196. https://doi.org/10.1007/s40314-021-01574-8 doi: 10.1007/s40314-021-01574-8
[19]	M. Saqib, M. Akram, S. Bashir, Certain efficient iterative methods for bipolar fuzzy system of linear equations, J. Intell. Fuzzy Syst., 39 (2020), 3971–3985. https://doi.org/10.3233/JIFS-200084 doi: 10.3233/JIFS-200084
[20]	M. Saqib, M. Akram, S. Bashir, T. Allahviranloo, Numerical solution of bipolar fuzzy initial value problem, J. Intell. Fuzzy Syst., 40 (2021), 1309–1341. https://doi.org/10.3233/JIFS-201619 doi: 10.3233/JIFS-201619
[21]	M. Saqib, M. Akram, S. Bashir, T. Allahviranloo, A Runge-Kutta numerical method to approximate the solution of bipolar fuzzy initial value problems, Comp. Appl. Math., 40 (2021), 151. https://doi.org/10.1007/s40314-021-01535-1 doi: 10.1007/s40314-021-01535-1
[22]	M. A. Mehmood, M. Akram, M. G. Alharbi, S. Bashir, Optimization of LR-type fully bipolar fuzzy linear programming problems, Math. Prob. Eng., 2021 (2021), 1199336. https://doi.org/10.1155/2021/1199336 doi: 10.1155/2021/1199336
[23]	M. A. Mehmood, M. Akram, M. G. Alharbi, S. Bashir, Solution of fully bipolar fuzzy linear programming models, Math. Probl. Eng., 2021 (2021), 996189. https://doi.org/10.1155/2021/9961891 doi: 10.1155/2021/9961891
[24]	A. Mahboob, G. Muhiuddin, A new type of fuzzy prime subset in ordered semigroups, New Math. Nat. Comput., 17 (2021), 739–752. https://doi.org/10.1142/S1793005721500368 doi: 10.1142/S1793005721500368
[25]	J. J. Chen, S. G. Li, S. Q. Ma, X. P. Wang, m-Polar fuzzy sets: An extension of bipolar fuzzy sets, Sci. World J., 2014 (2014), 416530. https://doi.org/10.1155/2014/416530 doi: 10.1155/2014/416530
[26]	M. Sarwar, M. Akram, New applications of m-polar fuzzy matroids, Symmetry, 9 (2017), 319. https://doi.org/10.3390/sym9120319 doi: 10.3390/sym9120319
[27]	A. Al-Masarwah, m-polar fuzzy ideals of BCK/BCI-algebras, J. King Saud Univ.-Sci., 31 (2019), 1220–1226. https://doi.org/10.1016/j.jksus.2018.10.002 doi: 10.1016/j.jksus.2018.10.002
[28]	A. N. A. S. Al-Masarwah, A. G. Ahmad, A new form of generalized m-PF ideals in BCK/BCI-algebras, Ann. Commun. Math., 2 (2019), 11–16.
[29]	A. Al-Masarwah, A. G. Ahmad, m-Polar (α, β)-fuzzy ideals in BCK/BCI-algebras, Symmetry, 11 (2019), 44. doi: https://doi.org/10.3390/sym11010044 doi: 10.3390/sym11010044
[30]	A. Al-Masarwah, On (complete) normality of m-PF subalgebras in BCK/BCI-algebras, AIMS Mathematics, 4 (2019), 740–750. doi: https://doi.org/10.3934/math.2019.3.740 doi: 10.3934/math.2019.3
[31]	G. Muhiuddin, M. M. Takallo, R. A. Borzooei, Y. B. Jun, m-polar fuzzy q-ideals in BCI-algebras, J. King Saud Univ. Sci., 32 (2020), 2803–2809. https://doi.org/10.1016/j.jksus.2020.07.001 doi: 10.1016/j.jksus.2020.07.001
[32]	G. Muhiuddin, D. Al-Kadi, A. Mahboob, A. Albjedi, Interval-valued m-polar fuzzy positive implicative ideals in BCK-algebras, Math. Probl. Eng., 2021 (2021), 1042091. https://doi.org/10.1155/2021/1042091 doi: 10.1155/2021/1042091
[33]	G. Muhiuddin, D. Al-Kadi, Interval valued m-polar fuzzy BCK/BCI-algebras, Int. J. Comput. Intell., 14 (2021), 1014–1021. https://doi.org/10.2991/ijcis.d.210223.003 doi: 10.2991/ijcis.d.210223.003
[34]	M. Shabir, M. Aslam, F. Pervaiz, m-polar fuzzy ideals in LA-semigroups, (submitted for publication).
[35]	S. Bashir, S. Shahzadi, A. N. Al-Kenani, M. Shabir, Regular and intra-regular semigroups in terms of m-polar fuzzy environment, Mathematic, 9 (2021), 2031. https://doi.org/10.3390/math9172031 doi: 10.3390/math9172031
[36]	M. Shabir, Y. Nawaz, M. Aslam, Semigroups characterized by the properties of their fuzzy ideals with thresholds, World Appl. Sci. J., 14 (2021), 1851–1865.
[37]	G. Mohanraj, M. Vela, On generalized fuzzy generalized fuzzy Bi-ideals of ternary semigroups, Bull. Int. Math. Virtual Inst., 9 (2019), 441–453. https://doi.org/10.7251/BIMVI1903441M doi: 10.7251/BIMVI1903441M
[38]	N. Rehman, M. Shabir, Characterizations of Ternary Semigroups by $\left({\alpha, \beta } \right)$-fuzzy Ideals, World Appl. Sci. J, 18 (2012), 1556–1570. https://doi.org/10.5829/idosi.wasj.2012.18.11.596 doi: 10.5829/idosi.wasj.2012.18.11.596
[39]	E. H. Hamouda, A study on anti fuzzy interior ideals of ternary semigroups, Asian J. Fuzzy Appl. Math., 2 (2014), 83–88.
[40]	D. T. Ho, J. M. Garibaldi, Context modelling in fuzzy systems, In: WCCI 2012 IEEE World Congress on Computational Intelligence, 2012, Brisbane, Australia.
[41]	V. R. Daddi, Intra-regular ternary semigroups, Glob. J. Pure Appl. Math., 13 (2017), 7689–7694.

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