Korean J. Math. Vol. 29 No. 1 (2021) pp.81-89
DOI: https://doi.org/10.11568/kjm.2021.29.1.81

Approximation operators and fuzzy rough sets in co-residuated lattices

Main Article Content

Ju-Mok Oh
Yong Chan Kim


In this paper, we introduce the notions of a distance function, Alexandrov topology and $\ominus$-upper ($\oplus$-lower) approximation operator based on complete co-residuated lattices. Under various relations, we define $(\oplus, \ominus)$-fuzzy rough set on complete co-residuated lattices. Moreover, we study their properties and give their examples.

Article Details

Supporting Agencies

Research Institute of Natural Science of Gangneung-Wonju National University.


[1] R. BVelohl avek, Fuzzy Relational Systems, Kluwer Academic Publishers, New York, 2002. Google Scholar

[2] P. Chen, D. Zhang, Alexandroff co-topological spaces, Fuzzy Sets and Systems, 161 (2010), 2505–2514. Google Scholar

[3] P. H ajek, Metamathematices of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998. Google Scholar

[4] U. H ̈ohle, E.P. Klement, Non-classical logic and their applications to fuzzy subsets, Kluwer Academic Publishers, Boston, 1995. Google Scholar

[5] U. H ̈ohle, S.E. Rodabaugh, Mathematics of Fuzzy Sets, Logic, Topology and Measure Theory, The Handbooks of Fuzzy Sets Series, Kluwer Academic Publishers, Dordrecht, 1999. Google Scholar

[6] F. Jinming, I-fuzzy Alexandrov topologies and specialization orders, Fuzzy Sets and Systems, 158 (2007), 2359–2374. Google Scholar

[7] Q. Junsheng, Hu. Bao Qing, On (⊙, &) -fuzzy rough sets based on residuated and co-residuated lattices, Fuzzy Sets and Systems, 336 (2018), 54–86. Google Scholar

[8] Y.C. Kim, Join-meet preserving maps and Alexandrov fuzzy topologies, Journal of Intelligent and Fuzzy Systems, 28 (2015), 457–467. Google Scholar

[9] Y.C. Kim,Categories of fuzzy preorders, approximation operators and Alexandrov topologies, Journal of Intelligent and Fuzzy Systems, 31 (2016), 1787–1793. Google Scholar

[10] Y.C. Kim, J.M Ko, Fuzzy complete lattices, Alexandrov L-fuzzy topologies and fuzzy rough sets, Journal of Intelligent and Fuzzy Systems, 38 (2020), 3253–3266. Google Scholar

[11] Y.C. Kim, J.M Ko, Preserving maps and approximation operators in complete co-residuated lattices, Journal of the korean Insitutute of Intelligent Systems, 30 (5)(2020), 389–398. Google Scholar

[12] H. Lai, D. Zhang, Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems, 157 (2006), 1865–1885. Google Scholar

[13] Z.M. Ma, B.Q. Hu, Topological and lattice structures of L-fuzzy rough set determined by lower and upper sets, Information Sciences, 218 (2013), 194–204. Google Scholar

[14] J.S. Mi, Y. Leung, H.Y. Zhao, T. Feng, Generalized fuzzy rough sets determined by a trianglar norm , Information Sciences, 178 (2008), 3203–3213. Google Scholar

[15] Z. Pawlak, Rough sets, Internat. J. Comput. Inform. Sci., 11 (1982), 341-356. Google Scholar

[16] Z. Pawlak, Rough sets: Theoretical Aspects of Reasoning about Data, System Theory, Knowledge Engineering and Problem Solving, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. Google Scholar

[17] A. M. Radzikowska, E.E. Kerre, A comparative study of fuzy rough sets, Fuzzy Sets and Systems, 126 (2002), 137–155. Google Scholar

[18] A.M. Radzikowska, E.E. Kerre, Characterisation of main classes of fuzzy relations using fuzzy modal operators, Fuzzy Sets and Systems, 152 (2005), 223–247. Google Scholar

[19] S.E. Rodabaugh, E.P. Klement, Topological and Algebraic Structures In Fuzzy Sets, The Hand-book of Recent Developments in the Mathematics of Fuzzy Sets, Kluwer Academic Publishers, Boston, Dordrecht, London, 2003. Google Scholar

[20] 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. Google Scholar

[21] S. P. Tiwari, A.K. Srivastava, Fuzzy rough sets, fuzzy preorders and fuzzy topologies, Fuzzy Sets and Systems, 210 (2013), 63–68. Google Scholar

[22] E. Turunen, Mathematics Behind Fuzzy Logic, A Springer-Verlag Co., 1999. Google Scholar

[23] M. Ward, R.P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc. 45 (1939), 335–354. Google Scholar

[24] W.Z. Wu, Y. Leung, J.S. Mi, On charterizations of (Φ, T )-fuzzy approximation operators, Fuzzy Sets and Systems, 154 (2005), 76–102. Google Scholar

[25] M.C. Zheng, G.J. Wang,Coresiduated lattice with applications, Fuzzy systems and Mathematics, 19 (2005), 1–6. Google Scholar