DOI QR코드

DOI QR Code

APPROXIMATION OPERATORS AND FUZZY ROUGH SETS IN CO-RESIDUATED LATTICES

  • Oh, Ju-Mok (Department of Mathematics, Gangneung-Wonju National) ;
  • Kim, Yong Chan (Department of Mathematics, Gangneung-Wonju National)
  • Received : 2020.12.01
  • Accepted : 2021.03.04
  • Published : 2021.03.30

Abstract

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

Keywords

References

  1. R. Belohlavek, Fuzzy Relational Systems, Kluwer Academic Publishers, New York, 2002.
  2. P. Chen, D. Zhang, Alexandroff co-topological spaces, Fuzzy Sets and Systems, 161 (2010), 2505-2514. https://doi.org/10.1016/j.fss.2010.01.002
  3. P. Hajek, Metamathematices of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998.
  4. U. Hohle, E.P. Klement, Non-classical logic and their applications to fuzzy subsets, Kluwer Academic Publishers, Boston, 1995.
  5. U. Hohle, S.E. Rodabaugh, Mathematics of Fuzzy Sets, Logic, Topology and Measure Theory, The Handbooks of Fuzzy Sets Series, Kluwer Academic Publishers, Dordrecht, 1999.
  6. F. Jinming, I-fuzzy Alexandrov topologies and specialization orders, Fuzzy Sets and Systems, 158 (2007), 2359-2374. https://doi.org/10.1016/j.fss.2007.05.001
  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. https://doi.org/10.1016/j.fss.2017.07.010
  8. Y.C. Kim, Join-meet preserving maps and Alexandrov fuzzy topologies, Journal of Intelligent and Fuzzy Systems, 28 (2015), 457-467. https://doi.org/10.3233/IFS-141322
  9. Y.C. Kim, Categories of fuzzy preorders, approximation operators and Alexandrov topologies, Journal of Intelligent and Fuzzy Systems, 31 (2016), 1787-1793. https://doi.org/10.3233/JIFS-152398
  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. https://doi.org/10.3233/jifs-191344
  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.
  12. H. Lai, D. Zhang, Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems, 157 (2006), 1865-1885. https://doi.org/10.1016/j.fss.2006.02.013
  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. https://doi.org/10.1016/j.ins.2012.06.029
  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. https://doi.org/10.1016/j.ins.2008.03.013
  15. Z. Pawlak, Rough sets, Internat. J. Comput. Inform. Sci., 11 (1982), 341-356. https://doi.org/10.1007/BF01001956
  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.
  17. A. M. Radzikowska, E.E. Kerre, A comparative study of fuzy rough sets, Fuzzy Sets and Systems, 126 (2002), 137-155. https://doi.org/10.1016/S0165-0114(01)00032-X
  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. https://doi.org/10.1016/j.fss.2004.09.005
  19. S.E. Rodabaugh, E.P. Klement, Topological and Algebraic Structures In Fuzzy Sets, The Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Kluwer Academic Publishers, Boston, Dordrecht, London, 2003.
  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. https://doi.org/10.1016/j.camwa.2009.03.100
  21. S. P. Tiwari, A.K. Srivastava, Fuzzy rough sets, fuzzy preorders and fuzzy topologies, Fuzzy Sets and Systems, 210 (2013), 63-68. https://doi.org/10.1016/j.fss.2012.06.001
  22. E. Turunen, Mathematics Behind Fuzzy Logic, A Springer-Verlag Co., 1999.
  23. M. Ward, R.P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc. 45 (1939), 335-354, https://doi.org/10.1090/S0002-9947-1939-1501995-3
  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. https://doi.org/10.1016/j.fss.2005.02.011
  25. M.C. Zheng, G.J. Wang, Coresiduated lattice with applications, Fuzzy systems and Mathematics, 19 (2005), 1-6.