A complete study on and-product of soft sets

Abstract

Soft set theory is a general mathematical framework for dealing with uncertainty. In this regard, soft set operations can be regarded as crucial concepts in soft set theory, since they offer new perspectives for dealing with issues containing parametric information. In this paper, we give a theoretical study on AND-product (∧-product), which is an essential concept in decision making problems, by investigating its whole algebraic properties in detail regarding soft F-subsets and soft M- equality, the strictest type of soft equality. Moreover, in order to complete some incomplete results concerning AND-product in the literature, we compare our properties by the formerly obtained properties regarding soft L-equality and soft J-equality. Furthermore, we handle the whole relations between AND-product and OR-product, the other keystone in decision making. Besides, by establishing some new results on distributive properties of AND-product over restricted, extended, and soft binary piecewise soft set operations, we prove that the set of all the soft sets over U together with restricted/extended union and AND-product is a commutative hemiring with identity as the set of all the soft sets over U together with restricted/extended symmetric difference and AND-product forms a commutative hemiring with identity in the sense of soft L-equality. As analyzing the algebraic structure of soft sets from the standpoint of operations gives profound insight into the potential uses of soft sets in classical and nonclassical logic and since theoretical foundations of soft computing approaches are derived from purely mathematical principles, this research will pave the way for a wide range of applications, including new decision-making approaches and innovative cryptography techniques based on soft sets.

Keywords

• AND-Product
• Hemiring
• Soft Equal Relations
• Soft Sets

References

1. REFERENCES
2. [1] Zadeh LA. Fuzzy sets. Inform Control 1965;8:338–353. [CrossRef]
3. [2] Molodtsov D. Soft set theory-First results. Comput Math Appl 1999;37:19–31. [CrossRef]
4. [3] Maji PK, Roy AR, Biswas R. An application of soft sets in a decision making problem. Comput Math Appl 2002;44:1077–1083. [CrossRef]
5. [4] Chen D, Tsang ECC, Yeung DS. Some notes on the parameterization reduction of soft sets. Int Conf Mach Learn Cyber 2003;3:1442–1445.
6. [5] Chen D, Tsang ECC, Yeung DS, Wang X. The parametrization reduction of soft sets and its applications. Comput Math Appl 2005;49;757–763. [CrossRef]
7. [6] Xiao Z, Chen L, Zhong B, Ye S. Recognition for soft information based on the theory of soft sets. In: Chen J (Ed.). IEEE Proceedings of ICSSSM-05 vol 2, pp 1104–1106, 2005. [CrossRef]
8. [7] Mushrif MM, Sengupta S, Ray AK. Texture Classification Using a Novel, Soft-Set Theory Based Classification Algorithm. In: Narayanan PJ, Nayar SK, Shum HY (Eds.). Computer Vision - ACCV 2006. ACCV 2006. Lecture Notes in Computer Science, vol 3851. Berlin, Heidelberg: Springer; 2006. [CrossRef]

9. [8] Herawan T, Deris MM. A direct proof of every rough set is a soft set. In: Third Asia ınternational Conference On Modelling and Simulation, Bandung, Bali, Indonesia, pp 119–124, 2009. [CrossRef]
10. [9] Herawan T, Deris MM. Soft Decision Making for Patients Suspected Influenza. In: Taniar D, Gervasi O, Murgante B, Pardede E, Apduhan BO (Eds.). Computational Science and Its Applications - ICCSA 2010. ICCSA 2010. Lecture Notes in Computer Science, vol 6018. Berlin, Heidelberg: Springer; 2010. [CrossRef]
11. [10] Herawan T. Soft set-based decision making for patients suspected influenza-like illness. Int J Mod Phys Conf Ser 2010;1:1–5.
12. [11] Çağman N, Enginoğlu S. Soft set theory and uni-int decision making, Eur J Oper Res 2010;207:848–855. [CrossRef]
13. [12] Çağman N, Enginoğlu S. Soft matrix theory and its decision making. Comput Math Appl 2010;59:3308–3314. [CrossRef]
14. [13] Gong K, Xiao Z, Zhang X. The bijective soft set with its operations. Comput Math Appl 2010;60:2270–2278. [CrossRef]
15. [14] Xiao Z, Gong K, Xia S, Zou Y. Exclusive disjunctive soft sets. Comput Math Appl 2010;59:2128–2137. [CrossRef] [15] Feng F, Li, Y, Çağman N. Generalized uni-int decision making schemes based on choice value soft sets. Eur J Oper Res 2012;220:162–170. [CrossRef]
16. [16] Feng Q, Zhou Y. Soft discernibility matrix and its applications in decision making. Appl Soft Comput 2014;24:749–756. [CrossRef]
17. [17] Kharal A. Soft approximations and uni-int decision making, Sci World J 2024;2024:327408.
18. [18] Dauda MK, Mamat M, Waziri MY. An application of soft set in decision making. J Teknol 2015;77:119–122. [CrossRef]
19. [19] Inthumathi V, Chitra V, Jayasree S. The role of operators on soft set in decision making problems. Int J Comput Appl Math 2017;12:899–910.
20. [20] Atagün AO, Kamacı H, Oktay O. Reduced soft matrices and generalized products with applications in decision making. Neural Comput Appl 2018;29:445–456. [CrossRef]
21. [21] Kamacı H, Saltık K, Akız HF, Atagün AO. Cardinality inverse soft matrix theory and its applications in multicriteria group decision making, J Intell Fuzzy Syst 2018;34:2031–2049. [CrossRef]
22. [22] Yang JL, Yao YY. Semantics of soft sets and three-way decision with soft sets. Knowl.-Based Syst 2020;194:105538. [CrossRef]
23. [23] Petchimuthu S, Garg H, Kamacı H, Atagün AO. The mean operators and generalized products of fuzzy soft matrices and their applications in MCGDM. Comput Appl Math 2020;39:1–32. [CrossRef]
24. [24] Zorlutuna I. Soft set-valued mappings and their application in decision making problems. Filomat 2021;35:1725–1733. [CrossRef]
25. [25] Maji PK, Biswas R, Roy AR. Soft set theory. Comput Math Appl 2003;45:555–562. [CrossRef]
26. [26] Pei D, Miao D. From sets to information systems. In: Hu X, Liu Q, Skowron A, Lin TY, Yager RR, Zhang B (Eds.). Proceedings of Granular Computing, IEEE 2, 617–621, 2005. [CrossRef]
27. [27] Yang CF. A note on: Soft set theory. Comput Math Appl 2008;45:555-562. [CrossRef]
28. [28] Ali MI, Feng F, Liu X, Min WK, Shabir M. On some new operations in soft set theory. Comput Math Appl 2009;57:1547–1553. [CrossRef]
29. [29] Feng F, Li YM, Davvaz B, Ali MI. Soft sets combined with fuzzy sets and rough sets: a tentative approach. Soft Comput 2010;14:899–911. [CrossRef]
30. [30] Jiang Y, Tang Y, Chen Q, Wang J, Tang S. Extending soft sets with description logics. Comput Math Appl 2010;59:2087–2096. [CrossRef]
31. [31] Ali MI, Shabir M, Naz M. Algebraic structures of soft sets associated with new operations. Comput Math Appl 2011;61:2647–2654. [CrossRef]
32. [32] Sezgin A, Atagün AO. On operations of soft sets. Comput Math Appl 2011;61:1457–1467. [CrossRef]
33. [33] Neog TJ, Sut DK. A new approach to the theory of soft sets. Int J Comput Appl 2011;32:1–6.
34. [34] Fu L. Notes on soft set operations. ARPN J Syst Softwares 2011;1:205–208.
35. [35] Ge X, Yang S. Investigations on some operations of soft sets. World Acad Sci Eng Technol 2011;75:1113–1116.
36. [36] Park JH, Kim OH, Kwun YC. Some properties of equivalence soft set relations. Comput Math Appl 2012;63:1079–1088. [CrossRef]
37. [37] Singh D, Onyeozili IA. Some conceptual misunderstanding of the fundamentals of soft set theory. ARPN J Syst Softw 2012;2:251–254.
38. [38] Singh D, Onyeozili IA. Some results on distributive and absorption properties on soft operations. IOSR J Math 2012;4:18–30. [CrossRef]
39. [39] Singh D, Onyeozili IA. On some new properties on soft sets operations, International J Comput Appl 2012;59:39–44. [CrossRef]
40. [40] Singh D, Onyeozili IA. Notes on soft matrices operations. ARPN J Sci Technol 2012;2:861–869.
41. [41] Zhu P, Wen Q. Operations on soft sets revisited. J Appl Math 2013;2013:105752. [CrossRef]
42. [42] Feng F, Li Y. Soft subsets and soft product operations. Inform Sci 2013;232:44–57. [CrossRef]
43. [43] Jayanta S. On algebraic structure of soft sets, Ann Fuzzy Math Inform Volume 2014;7:1013–1020.
44. [44] Onyeozili IA, Gwary TM. A study of the fundamentals of soft set theory, International J Sci Technol 2014;3:132–143.
45. [45] Husain S, Shivani K. A study of properties of soft set and ıts applications. Int Res J Eng Technol 2018;5:363–372.
46. [46] Eren ÖF, Çalışıcı H. On some operations of soft sets, The Fourth International Conference on Computational Mathematics and Engineering Sciences, Antalya, 2019.
47. [47] Sezgin A, Shahzad A, Mehmood A. new operation on soft sets: extended difference of soft sets. J New Theory 2019;27:33–42.
48. [48] Stojanovic NS. A new operation on soft sets: extended symmetric difference of soft sets. Military Technical Courier 2021;69:779–791. [CrossRef] [49] Yavuz E. Soft binary piecewise operations and their properties. Amasya: Amasya University, The Graduate School of Natural and Applied Sciences Master of Science in Mathematics Department, 2024.
49. [50] Qin KY, Hong ZY. On soft equality. J Comput Appl Math 2010;234:1347–1355. [CrossRef] [51] Jun YB, Yang X. A note on the paper "Combination of interval-valued fuzzy set and soft set. Comput Math Appl 2011;61:1468–1470. [CrossRef]
50. [52] Liu XY, Feng F, Jun YB. A note on generalized soft equal relations, Computers and Mathematics with Applications, 2012;64:572–578. [CrossRef]
51. [53] Feng F, Yongming L. Soft subsets and soft product operations. Inform Sci 2013;232:44–57. [CrossRef]
52. [54] Abbas M, Ali B, Romaguer S. On generalized soft equality and soft lattice structure. Filomat 2014;28:1191–1203. [CrossRef]
53. [55] Abbas M, Ali MI, Romaguera S. Generalized operations in soft set theory via relaxed conditions on parameters, Filomat 2017;31:5955–5964. [CrossRef]
54. [56] Al-shami TM. Investigation and corrigendum to some results related to g-soft equality and g f -soft equality relations. Filomat 2019;33:3375–3383. [CrossRef]
55. [57] Alshasi T, El-Shafei T. T-soft equality relation. Turk J Math 2020;44:25. [CrossRef]
56. [58] Ali B, Saleem N, Sundus N, Khaleeq S, Saeed M, George RA. Contribution to the theory of soft sets via generalized relaxed operations. Mathematics 2022;10:26–36. [CrossRef]
57. [59] Vandiver HS. Note on a simple type of algebra in which the cancellation law of addition does not hold. Bullet Am Math Soc 1934;40:914–920. [CrossRef]
58. [60] Vasanthi T, Sulochana N. On the additive and multiplicative structure of semirings. Ann Pure Appl Math 2013;3:78–84.
59. [61] Kaya A, Satyanarayana M. Semirings satisfying properties of distributive type. Proceed Am Math Soc 1981;82:341–346. [CrossRef]
60. [62] Karvellas PH. Inversive semirings. J Australian Math Soc 1974;18:277–288. [CrossRef]
61. [63] Goodearl KR. Von Neumann Regular Rings. London: Pitman; 1979.
62. [64] Petrich M. Introduction to Semiring: Ohio: Charles E Merrill Publishing Company; 1973
63. [65] Reutenauer C, Straubing H. Inversion of matrices over a commutative semiring. J Algebra 1984;88:350–360. [CrossRef]
64. [66] Glazek K. A guide to litrature on semirings and their applications in mathematics and information sciences: with complete bibliography. Nederland: Kluwer Acad. Publ; 2012.
65. [67] Kolokoltsov VN, Maslov VP. Idempotent analysis and its applications, in: Mathematics and its Applications. Nederland: Kluwer Acad. Publ; 1997. [CrossRef] [68] Hopcroft JE, Ullman JD. Introduction to automata theory, languages and computation, Reading, MA: Addison Wesley; 1979.
66. [69] Beasley LB, Pullman NG. Operators that preserves semiring matrix functions, Linear Algebra Appl 1988;99:199–216. [CrossRef]
67. [70] Beasley LB, Pullman NG. Linear operators strongly preserving idempotent matrices over semirings. Linear Algebra Appl 1992;160:217–229. [CrossRef]
68. [71] Ghosh S. Matrices over semirings. Inform Sci 1996;90:221–230. [CrossRef]
69. [72] Wechler W. The concept of fuzziness in automata and language theory. Verlag, Berlin: Akademic; 1978. [CrossRef]
70. [73] Golan JS. Semirings and their applications. Nederland: Kluwer Acad Publication; 1999. [CrossRef]
71. [74] Hebisch U, Weinert HJ. Semirings: algebraic theory and applications in the computer science. Singapore: World Scientific; 1998. [CrossRef]
72. [75] Mordeson JN, Malik DS. Fuzzy automata and languages, theory and applications, in: computational mathematics series. Boca Raton: Chapman and Hall, CRC; 2002.
73. [76] Pant S, Dagtoros K, Kholil MI, Vivas A. Matrices: Peculiar determinant property. OPS 2024;1:1–7.