On L-fuzzy approximation operators and L-fuzzy relations on residuated lattices

Document Type : Original Article


Department of Mathematics, Tokyo Denki University, Tokyo, Japan


We consider properties of L-fuzzy relations and L-normal operators for a residuated lattice L in detail and show that the class RL(U) of all L-fuzzy relations on U and the class NL(U) of all L-normal operators are residuated lattices and they are isomorphic as lattices. Moreover, we prove that for any L-normal operators F, it is reflexive (or transitive) if and only if the L-fuzzy relation RF induced by F is reflexive (or transitive) respectively.


[1] M.E. Abd El-Monsef, A.M. Kozae, M.K. El-Bably, On generalizing covering approximation space, Journal of the Egyptian Mathematical Society, 23 (2015), 535–545.
[2] N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated lattices: An algebraic glimpse at substructural logics, Studies in  Logic and the Foundations of Mathematics, 151 (2007), Elsevier. 
[3] D. Génény, S. Radeleczki, Rough L-fuzzy sets: Their representation and related structures, International Journal of  Approximate Reasoning, 142 (2022), 1–12.
[4] M. Kondo, On the structure of generalized rough sets, Information Sciences, 176 (2005), 589–600.
[5] M. Kondo, On topologies defined by binary relations in rough sets, IJCRS 2019, Lecture Notes in Computer Sciences  LNCS (Springer), 11499 (2019), 66–77.
[6] Z. Li, Topological properties of generalized rough sets, FKSD 2010 Seventh International Conference on Fuzzy and  Knowledge Discovery, (2010), 2067–2070.
[7] S. Mahato, S.P. Tiwari, On fuzzy approximation operators and fuzzy transformation systems, 11th Conference of the  European Society for Fuzzy Logic and Technology (EUSFLAT 2019), (2019), 274–280.
[8] B. Pang, J.S. Mi, W. Yao, L-fuzzy rough approximation operators via three new types of L-fuzzy relations, Soft  Computing, 23 (2019), 11433–11446.
[9] Z. Pawlak, Rough sets, International Journal of Computer Science and Information, 11 (1982), 341–356.
[10] A.M. Radzikowska, E.E. Kerre, Fuzzy rough sets based on residuated lattices, Transactions on Rough Sets II, LNCS 3135 (Springer), (2004), 278–296.
[11] Y.H. She, G.J. Wang, An axiomatic approach of fuzzy rough sets based on residuated lattices, Computers and  Mathematics with Applications, 58 (2009), 189–201.
[12] P. Wang, Q. Li, The characterizations of upper approximation operators based on coverings, Soft Computing, 23  (2019), 3217–3228.
[13] W. Zhu, Generalized rough sets based on relations, Information Sciences, 177 (2007), 1499–1508.
[14] J. Zeng, P. Wang, L-fuzzy approximating spaces, Journal of Intelligent and Fuzzy Systems, 37 (2019), 5031–5038.