Abstract

This paper presents a comparative study of three important composition operators in fuzzy relations, such as max-min, max-product, and max-average compositions. The study demonstrates that these operators hold for a reflexive fuzzy relation. Furthermore, an illustrative example is provided for each type of composition, using matrices of order3×3.

Keywords

Composition of fuzzy relations, Fuzzy relations, Fuzzy sets, Reflexive, Order of matrix.

References

[1] S. K. Pundir and R., Pundir, Fuzzy Sets and Their Applications, Pragati Prakashan, Tenth Edition, 2022.
[Publisher Link]

[2] M.A.  Shakhatreh, and T.A. Qawasmeh, “Associativity of Max-Min Composition of Three  Fuzzy Relations,” Italian Journal of Pure and Applied Mathematics, vol. 44, pp. 224-228, 2020.
[Google Scholar] [Publisher Link]

[3] L.A. Zadeh, “Fuzzy Sets,” Information and Control, vol. 8, no. 3, pp. 338-353, 1965.
[CrossRef] [Google Scholar] [Publisher Link]

[4] R.E. Bellman, and L.A. Zadeh. “Decision-Making in a Fuzzy Environment,” Management Science, vol. 17, no. 4, pp. 141-164, 1970.
[CrossRef] [Google Scholar] [Publisher Link]

[5] E.H. Mamdani, “Application of Fuzzy Algorithms for Control of Simple Dynamic Plant,” Proceedings of the IEE, vol. 121, no. 12, pp.1585-1588, 1974.
[CrossRef] [Google Scholar] [Publisher Link]

[6] James C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Springer, 1981.
[CrossRef] [Google Scholar] [Publisher Link]

[7] Fang-Fang Guo, Rong Fu, and Jie Shen, “Inverses of Fuzzy Relation Matrices with Addition-Min Composition,” Fuzzy Sets and Systems, vol. 490, 2024.
[CrossRef] [Google Scholar] [Publisher Link]

[8] H.J. Zimmermann, Fuzzy Set Theory and Its Applications, Kluwer Academic Publishers, 1991.
[Google Scholar] [Publisher Link]