Abstract
The multi-fuzzy soft set theory has recently been introduced and it has started to be applied in some fields such as decision making and medical diagnosis. In this paper, algebraic structure of multi-fuzzy soft sets is studied. Several related properties of some operations on multi-fuzzy soft sets are investigated. Two lattice structures of multi-fuzzy soft sets are constructed. It is shown that these lattices are distributive and whence modular. Additionally, the ordering relations on the lattices of multi-fuzzy soft sets are presented. Moreover, by giving an example, it is indicated that some pairs of operations on multi-fuzzy soft sets do not satisfy the absorption rule which is necessary to form a lattice. So it is proved that a lattice structure cannot be constructed by using these operations.
Keywords
References
- Akın C. 2021. Multi-fuzzy soft groups. Soft Comput, 25: 137-145.
- Aygünoğlu A, Aygün H. 2009. Introduction to fuzzy soft groups. Comput Math Appl, 58: 1279-1286.
- Birkhoff G. 1984. Lattice theory. American Math Soc, Providence, US.
- Feng F, Jun YB, Zhao XZ. 2008. Soft semirings. Comput Math Appl, 56: 2621-2628.
- Gratzer G. 1978. General lattice theory. Birkhauser Verlag, Basel, Switzerland.
- Jun YB. 2008. Soft BCK/BCI-algebras. Comput Math Appl, 56: 1408-1413.
- Kazancı O, Mayerova S, Davvaz B. 2022. Algebraic hyperstructure of multi-fuzzy soft sets related to polygroups. Mathematics, 10: 2178.
- Kazancı O, Yılmaz Ş, Yamak S. 2010. Soft sets and soft BCH-algebras. Hacettepe J Math Stat, 39(2): 205 – 217.
- Maji PK, Biswas R, Roy AR. 2001. Fuzzy soft sets. J Fuzzy Math, 9: 589-602.
- Majumdar P, Samanta SK. 2010. Generalised fuzzy soft sets. Comput Math Appl, 59: 1425-1432.
- Molodtsov D. 1999. Soft set theory-first results. Comput Math Appl, 37: 19-31.
- Pawlak Z. 1982. Rough sets. Int J Info Comput Sci, 11: 341-356.
- Qin K, Hong Z. 2010. On soft equality. J Comput Appl Math, 234: 1347-1355.
- Sebastian S, Ramakrishnan TV. 2011a. Multi-fuzzy sets: an extension of fuzzy sets. Fuzzy Info Eng, 1: 35-43.
- Sebastian S, Ramakrishnan TV. 2011b. Multi-fuzzy subgroups. Int J Contemp Math Sci, 8: 365-372.
- Shao Y, Qin K. 2012. Fuzzy soft sets and fuzzy soft lattices. Int J Comput Intel Syst, 5 (6): 1135-1147.
- Skornjakov LA. 1977. Elements of lattice theory. Hindustan Publishing Corporation, New Delhi, India.
- Sun QM, Zhang ZL, Liu J. 2008. Soft sets and soft modules. Lecture Notes Comput Sci, 5009: 403-409.
- Yang C. 2011. Fuzzy soft semigroups and fuzzy soft ideals. Comput Math Appl, 61: 255-261.
- Yang Y, Tan X, Meng C. 2013. The multi-fuzzy soft set and its application in decision making. Appl Math Model, 37: 4915-4923.
- Zadeh LA. 1965. Fuzzy sets. Info Cont, 8: 338-353.
Abstract
The multi-fuzzy soft set theory has recently been introduced and it has started to be applied in some fields such as decision making and medical diagnosis. In this paper, algebraic structure of multi-fuzzy soft sets is studied. Several related properties of some operations on multi-fuzzy soft sets are investigated. Two lattice structures of multi-fuzzy soft sets are constructed. It is shown that these lattices are distributive and whence modular. Additionally, the ordering relations on the lattices of multi-fuzzy soft sets are presented. Moreover, by giving an example, it is indicated that some pairs of operations on multi-fuzzy soft sets do not satisfy the absorption rule which is necessary to form a lattice. So it is proved that a lattice structure cannot be constructed by using these operations.
Keywords
References
- Akın C. 2021. Multi-fuzzy soft groups. Soft Comput, 25: 137-145.
- Aygünoğlu A, Aygün H. 2009. Introduction to fuzzy soft groups. Comput Math Appl, 58: 1279-1286.
- Birkhoff G. 1984. Lattice theory. American Math Soc, Providence, US.
- Feng F, Jun YB, Zhao XZ. 2008. Soft semirings. Comput Math Appl, 56: 2621-2628.
- Gratzer G. 1978. General lattice theory. Birkhauser Verlag, Basel, Switzerland.
- Jun YB. 2008. Soft BCK/BCI-algebras. Comput Math Appl, 56: 1408-1413.
- Kazancı O, Mayerova S, Davvaz B. 2022. Algebraic hyperstructure of multi-fuzzy soft sets related to polygroups. Mathematics, 10: 2178.
- Kazancı O, Yılmaz Ş, Yamak S. 2010. Soft sets and soft BCH-algebras. Hacettepe J Math Stat, 39(2): 205 – 217.
- Maji PK, Biswas R, Roy AR. 2001. Fuzzy soft sets. J Fuzzy Math, 9: 589-602.
- Majumdar P, Samanta SK. 2010. Generalised fuzzy soft sets. Comput Math Appl, 59: 1425-1432.
- Molodtsov D. 1999. Soft set theory-first results. Comput Math Appl, 37: 19-31.
- Pawlak Z. 1982. Rough sets. Int J Info Comput Sci, 11: 341-356.
- Qin K, Hong Z. 2010. On soft equality. J Comput Appl Math, 234: 1347-1355.
- Sebastian S, Ramakrishnan TV. 2011a. Multi-fuzzy sets: an extension of fuzzy sets. Fuzzy Info Eng, 1: 35-43.
- Sebastian S, Ramakrishnan TV. 2011b. Multi-fuzzy subgroups. Int J Contemp Math Sci, 8: 365-372.
- Shao Y, Qin K. 2012. Fuzzy soft sets and fuzzy soft lattices. Int J Comput Intel Syst, 5 (6): 1135-1147.
- Skornjakov LA. 1977. Elements of lattice theory. Hindustan Publishing Corporation, New Delhi, India.
- Sun QM, Zhang ZL, Liu J. 2008. Soft sets and soft modules. Lecture Notes Comput Sci, 5009: 403-409.
- Yang C. 2011. Fuzzy soft semigroups and fuzzy soft ideals. Comput Math Appl, 61: 255-261.
- Yang Y, Tan X, Meng C. 2013. The multi-fuzzy soft set and its application in decision making. Appl Math Model, 37: 4915-4923.
- Zadeh LA. 1965. Fuzzy sets. Info Cont, 8: 338-353.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Logic, Set Theory, Lattices and Universal Algebra |
| Journal Section | Research Articles |
| Authors | |
| Publication Date | January 15, 2025 |
| Submission Date | November 8, 2024 |
| Acceptance Date | December 12, 2024 |
| Published in Issue | Year 2025 |