Roughness of soft sets and fuzzy sets in semigroups based on set-valued picture hesitant fuzzy relations

Rukchart Prasertpong; Rukchart Prasertpong

doi:10.3934/math.2022160

AIMS Mathematics

2022, Volume 7, Issue 2: 2891-2928. doi: 10.3934/math.2022160

Previous Article Next Article

Research article

Roughness of soft sets and fuzzy sets in semigroups based on set-valued picture hesitant fuzzy relations

Rukchart Prasertpong ^,

Division of Mathematics and Statistics, Faculty of Science and Technology, Nakhon Sawan Rajabhat University, Nakhon Sawan 60000, Thailand

Received: 23 July 2021 Accepted: 02 November 2021 Published: 22 November 2021
MSC : 03E72, 03G25, 08A72

In the philosophy of rough set theory, the methodologies of rough soft sets and rough fuzzy sets have been being examined to be efficient mathematical tools to deal with unpredictability. The basic of approximations in rough set theory is based on equivalence relations. In the aftermath, such theory is extended by arbitrary binary relations and fuzzy relations for more wide approximation spaces. In recent years, the notion of picture hesitant fuzzy relations by Mathew et al. can be considered as a novel extension of fuzzy relations. Then this paper proposes extended approximations into rough soft sets and rough fuzzy sets from the viewpoint of its. We give corresponding examples to illustrate the correctness of such approximations. The relationships between the set-valued picture hesitant fuzzy relations with the upper (resp., lower) rough approximations of soft sets and fuzzy sets are investigated. Especially, it is shown that every non-rough soft set and non-rough fuzzy set can be induced by set-valued picture hesitant fuzzy reflexive relations and set-valued picture hesitant fuzzy antisymmetric relations. By processing the approximations and advantages in the new existing tools, some terms and products have been applied to semigroups. Then, we provide attractive results of upper (resp., lower) rough approximations of prime idealistic soft semigroups over semigroups and fuzzy prime ideals of semigroups induced by set-valued picture hesitant fuzzy relations on semigroups.
- rough set,
- rough soft set,
- rough fuzzy set,
- prime idealistic soft semigroup over semigroup,
- fuzzy prime ideal of semigroup,
- set-valued picture hesitant fuzzy relation
Citation: Rukchart Prasertpong. Roughness of soft sets and fuzzy sets in semigroups based on set-valued picture hesitant fuzzy relations[J]. AIMS Mathematics, 2022, 7(2): 2891-2928. doi: 10.3934/math.2022160

Related Papers:

Abstract

In the philosophy of rough set theory, the methodologies of rough soft sets and rough fuzzy sets have been being examined to be efficient mathematical tools to deal with unpredictability. The basic of approximations in rough set theory is based on equivalence relations. In the aftermath, such theory is extended by arbitrary binary relations and fuzzy relations for more wide approximation spaces. In recent years, the notion of picture hesitant fuzzy relations by Mathew et al. can be considered as a novel extension of fuzzy relations. Then this paper proposes extended approximations into rough soft sets and rough fuzzy sets from the viewpoint of its. We give corresponding examples to illustrate the correctness of such approximations. The relationships between the set-valued picture hesitant fuzzy relations with the upper (resp., lower) rough approximations of soft sets and fuzzy sets are investigated. Especially, it is shown that every non-rough soft set and non-rough fuzzy set can be induced by set-valued picture hesitant fuzzy reflexive relations and set-valued picture hesitant fuzzy antisymmetric relations. By processing the approximations and advantages in the new existing tools, some terms and products have been applied to semigroups. Then, we provide attractive results of upper (resp., lower) rough approximations of prime idealistic soft semigroups over semigroups and fuzzy prime ideals of semigroups induced by set-valued picture hesitant fuzzy relations on semigroups.

References

[1]	G. Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre, Leipzig: Teubner, 1883.
[2]	Z. Pawlak, A. Skowron, Rudiments of rough sets, Inform. Sci., 177 (2007), 3–27. doi: 10.1016/j.ins.2006.06.003.
[3]	Z. Pawlak, Rough sets, Int. J. Comput. Inf. Sci., 11 (1982), 341–356. doi: 10.1007/BF01001956.
[4]	L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353.
[5]	D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, Int. J. Gen. Syst., 17 (1990), 191–209. doi: 10.1080/03081079008935107. doi: 10.1080/03081079008935107
[6]	D. Molodtsov, Soft set theory-first results, Comput. Math. Appl., 37 (1999), 19–31. doi: 10.1016/S0898-1221(99)00056-5. doi: 10.1016/S0898-1221(99)00056-5
[7]	F. Feng, C. X. Li, B. Davvaz, M. I. Ali, Soft sets combined with fuzzy sets and rough sets: A tentative approach, Soft Comput., 14 (2010), 899–911. doi: 10.1007/s00500-009-0465-6. doi: 10.1007/s00500-009-0465-6
[8]	Y. Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Inform. Sci., 111 (1998), 239–259. doi: 10.1016/S0020-0255(98)10006-3. doi: 10.1016/S0020-0255(98)10006-3
[9]	R. Mareay, Generalized rough sets based on neighborhood systems and topological spaces, J. Egypt. Math. Soc., 24 (2016), 603–608. doi: 10.1016/j.joems.2016.02.002. doi: 10.1016/j.joems.2016.02.002
[10]	R. Prasertpong, M. Siripitukdet, On rough sets induced by fuzzy relations approach in semigroups, Open Math., 16 (2018), 1634–1650. doi: 10.1515/math-2018-0136. doi: 10.1515/math-2018-0136
[11]	R. Prasertpong, M. Siripitukdet, Rough set models induced by serial fuzzy relations approach in semigroups, Eng. Let., 27 (2019), 216–225.
[12]	R. Prasertpong, M. Siripitukdet, Generalizations of rough sets induced by binary relations approach in semigroups, J. Intell. Fuzzy Syst., 36 (2019), 5583–5596. doi: 10.3233/JIFS-181435. doi: 10.3233/JIFS-181435
[13]	R. Prasertpong, M. Siripitukdet, Applying generalized rough set concepts to approximation spaces of semigroups, IAENG Int. J. Appl. Math., 49 (2019), 51–60.
[14]	L. A. Zadeh, Similarity relations and fuzzy orderings, Inform. Sci., 3 (1971), 117–200. doi: 10.1016/S0020-0255(71)80005-1. doi: 10.1016/S0020-0255(71)80005-1
[15]	K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), 87–96. doi: 10.1016/S0165-0114(86)80034-3.
[16]	P. J. Burillo, H. Bustince, Intuitionistic fuzzy relations (Part I), Mathware Soft Comput., 2 (1995), 5–38.
[17]	F. Smarandache, A unifying field in logics: Neutrosophic logic. Neutrosophy set, neutrosophic probability and statistics, 4 Eds., Rehoboth: American Research Press, 2005.
[18]	B. C. Cuong, Picture fuzzy sets, J. Comput. Sci. Cybern., 30 (2014), 409–420. doi: 10.15625/1813-9663/30/4/5032.
[19]	V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst., 25 (2010), 529–539. doi: 10.1002/int.20418.
[20]	B. Zhu, Studies on consistency measure of hesitant fuzzy preference relations, Procedia Comput. Sci., 17 (2013), 457–464. doi: 10.1016/j.procs.2013.05.059. doi: 10.1016/j.procs.2013.05.059
[21]	R. Wang, B. Shuai, Z. S. Chen, K. S. Chin, J. H. Zhu, Revisiting the role of hesitant multiplicative preference relations in group decision making with novel consistency improving and consensus reaching processes, Int. J. Comput. Intell. Syst., 12 (2019), 1029–1046. doi: 10.2991/ijcis.d.190823.001. doi: 10.2991/ijcis.d.190823.001
[22]	Z. S. Chen, X. Zhang, W. Pedrycz, X. J. Wang, K. S. Chin, L Martínezf, $K$-means clustering for the aggregation of HFLTS possibility distributions: $N$-two-stage algorithmic paradigm, Knowl.-Based Syst., 227 (2021), 107230. doi: 10.1016/j.knosys.2021.107230.
[23]	Z. M. Zhang, S. M. Chen, Group decision making based on acceptable multiplicative consistency and consensus of hesitant fuzzy linguistic preference relations, Inform. Sci., 541 (2020), 531–550. doi: 10.1016/j.ins.2020.07.024. doi: 10.1016/j.ins.2020.07.024
[24]	R. Wang, Y. L. Li, Picture hesitant fuzzy set and its application to multiple criteria decision-making, Symmetry, 10 (2018), 1–29. doi: 10.3390/sym10070295. doi: 10.3390/sym10070295
[25]	B. Mathew, S. J. John, J. C. R. Alcantud, Multi-granulation picture hesitant fuzzy rough sets, Symmetry, 12 (2020), 1–17. doi: 10.3390/sym12030362. doi: 10.3390/sym12030362
[26]	T. Y. Lin, N. Cercone, Rough sets and data mining, 1 Ed., Boston: Springer, 1997. doi: 10.1007/978-1-4613-1461-5.
[27]	Q. H. Zhang, Q. Xie, G. Y. Wang, A survey on rough set theory and its applications, CAAI T. Intell. Techno., 1 (2016), 323–333. doi: 10.1016/j.trit.2016.11.001. doi: 10.1016/j.trit.2016.11.001
[28]	Q. M. Xiao, Z. L. Zhang, Rough prime ideals and rough fuzzy prime ideals in semigroups, Inform. Sci., 176 (2006), 725–733. doi: 10.1016/j.ins.2004.12.010. doi: 10.1016/j.ins.2004.12.010
[29]	O. Kazancı, B. Davvaz, On the structure of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in commutative rings, Inform. Sci., 178 (2008), 1343–1354. doi: 10.1016/j.ins.2007.10.005. doi: 10.1016/j.ins.2007.10.005
[30]	Q. J. Luo, G. J. Wang, Roughness and fuzziness in quantales, Inform. Sci., 271 (2014), 14–30. doi: 10.1016/j.ins.2014.02.105. doi: 10.1016/j.ins.2014.02.105
[31]	J. M. Zhan, Q. Liu, B. Davvaz, A new rough set theory: Rough soft hemirings, J. Intell. Fuzzy Syst., 28 (2015), 1687–1697. doi: 10.3233/IFS-141455. doi: 10.3233/IFS-141455
[32]	S. K. Roy, S. Bera, Approximation of rough soft set and its application to lattice, Fuzzy Inf. Eng., 7 (2015), 379–387. doi: 10.1016/j.fiae.2015.09.008. doi: 10.1016/j.fiae.2015.09.008
[33]	W. J. Pan, J. M. Zhan, Rough fuzzy groups and rough soft groups, Ital. J. Pure Appl. Math., 36 (2016), 617–628.
[34]	J. M. Zhan, B. Davvaz, A kind of new rough set: Rough soft sets and rough soft rings, J. Intell. Fuzzy Syst., 30 (2016), 475–483. doi: 10.3233/IFS-151772. doi: 10.3233/IFS-151772
[35]	Q. M. Wang, J. M. Zhan, Rough semigroups and rough fuzzy semigroups based on fuzzy ideals, Open Math., 14 (2016), 1114–1121. doi: 10.1515/math-2016-0102. doi: 10.1515/math-2016-0102
[36]	Q. M. Wang, J. M. Zhan, A novel view of rough soft semigroups based on fuzzy ideals, Ital. J. Pure Appl. Math., 37 (2017), 673–686.
[37]	J. M. Zhan, X. W. Zhou, D. J. Xiang, Rough soft n-ary semigroups based on a novel congruence relation and corresponding decision making, J. Intell. Fuzzy Syst., 33 (2017), 693–703. doi: 10.3233/JIFS-161497. doi: 10.3233/JIFS-161497
[38]	J. M. Zhan, Q. Liu, W. Zhu, Another approach to rough soft hemirings and corresponding decision making, Soft Comput., 21 (2017), 3769–3780. doi: 10.1007/s00500-016-2058-5. doi: 10.1007/s00500-016-2058-5
[39]	S. M. Qurashi, M. Shabir, Generalized rough fuzzy ideals in quantales, Discrete Dyn. Nat. Soc., 2018 (2018), 1–11. doi: 10.1155/2018/1085201. doi: 10.1155/2018/1085201
[40]	J. C. R. Alcantud, F. Feng, R. R. Yager, An $N$-soft set approach to rough sets, IEEE Trans. Fuzzy Syst., 28 (2019), 2996–3007. doi: 10.1109/TFUZZ.2019.2946526. doi: 10.1109/TFUZZ.2019.2946526
[41]	R. S. Kanwal, M. Shabir, Rough approximation of a fuzzy set in semigroups based on soft relations, Comp. Appl. Math., 38 (2019), 1–23. doi: 10.1007/s40314-019-0851-3. doi: 10.1007/s40314-019-0851-3
[42]	A. Hussain, T. Mahmood, M. I. Ali, Rough Pythagorean fuzzy ideals in semigroups, Comp. Appl. Math., 38 (2019), 1–15. doi: 10.1007/s40314-019-0824-6. doi: 10.1007/s40314-019-0824-6
[43]	R. Chinram, T. Panityakul, Rough Pythagorean fuzzy ideals in ternary semigroups, J. Math. Comput. Sci., 20 (2020), 302–312. doi: 10.22436/jmcs.020.04.04. doi: 10.22436/jmcs.020.04.04
[44]	A. Satirad, R. Chinram, A. Iampan, Pythagorean fuzzy sets in UP-algebras and approximations, AIMS Math., 6 (2021), 6002–6032. doi: 10.3934/math.2021354. doi: 10.3934/math.2021354
[45]	A. Elmoasry, On rough fuzzy prime ideals in left almost semigroups, Int. J. Anal. Appl., 19 (2021), 455–464. doi: 10.28924/2291-8639-19-2021-455. doi: 10.28924/2291-8639-19-2021-455
[46]	A. H. Clifford, G. B. Preston, The algebraic theory of semigroups, Providence, RI: American Mathematical Society, 1961. doi: 10.1090/surv/007.1.
[47]	J. M. Howie, Fundamentals of semigroup theory, United States: Oxford University Press, 1995.
[48]	Š. Schwarz, Prime ideals and maximal ideals in semigroups, Czechoslovak Math. J., 19 (1969), 72–79.
[49]	J. N. Mordeson, D. S. Malik, N. Kuroki, Fuzzy semigroups, Berlin, Heidelberg: Springer, 2003. doi: 10.1007/978-3-540-37125-0.
[50]	M. K. Chakraborty, M. Das, On fuzzy equivalence-I, Fuzzy Sets Syst., 11 (1983), 185–193.
[51]	M. K. Chakraborty, M. Das, On fuzzy equivalence-II, Fuzzy Sets Syst., 11 (1983), 299–307.
[52]	M. K. Chakraborty, S. Sarkar, Fuzzy antisymmetry and order, Fuzzy Sets Syst., 21 (1987), 169–182. doi: 10.1016/0165-0114(87)90162-X. doi: 10.1016/0165-0114(87)90162-X
[53]	P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Comput. Math. Appl., 45 (2003), 555–562. doi: 10.1016/S0898-1221(03)00016-6.
[54]	M. I. Ali, F. Feng, X. Y. Liu, 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. doi: 10.1016/j.camwa.2008.11.009
[55]	M. I. Ali, M. Shabir, K. P. Shum, On soft ideals over semigroups, Southeast Asian Bull. Math., 34 (2010), 595–610.
[56]	F. Feng, M. I. Ali, M. Shabir, Soft relations applied to semigroups, Filomat, 27 (2013), 1183–1196. doi: 10.2298/FIL1307183F. doi: 10.2298/FIL1307183F

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)