Soft Computing

, Volume 22, Issue 12, pp 3843–3855 | Cite as

On the measure of M-rough approximation of L-fuzzy sets

  • Sang-Eon Han
  • Alexander ŠostakEmail author


We develop an approach allowing to measure the “quality” of rough approximation of fuzzy sets. It is based on what we call “an approximative quadruple” \(Q=(L,M,\varphi ,\psi )\) where L and M are complete lattice commutative monoids and \(\varphi : L \rightarrow M\), \(\psi : M \rightarrow L\) are mappings satisfying certain conditions. By realization of this scheme, we get measures of upper and lower rough approximation for L-fuzzy subsets of a set equipped with an M-preoder \(R: X\times X \rightarrow M\). In case R is symmetric, these measures coincide. Basic properties of such measures are studied. Besides, we present an interpretation of measures of rough approximation in terms of LM-fuzzy topologies.


L-fuzzy set Upper M-rough approximation operator Lower M-rough approximation operator Measure of inclusion Measure of M-rough approximation of an L-fuzzy set Ditopology LM-ditopology 



The first named author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2016R1D1A3A03918403). The second named author expresses gratefulness to Chonbuk National University and KIAS (Korea Institute for Advanced Study) for the financial support of his visits to Chonbuk National university in years 2012, 2014 and 2016 during which an essential part of this research was done. Both authors are grateful to the anonymous referee for reading the paper carefully and making valuable comments which allowed to eliminate some mistakes and to improve the exposition of the material.

Compliance with ethical standards

Conflict of interest

The authors confirm that they do not have conflict of interests.


  1. Brown LM, Ertürk R, Dost Ş (2000) Ditopological texture spaces and fuzzy topology, I. Basic concepts. Fuzzy Sets Syst 110:227–236CrossRefzbMATHGoogle Scholar
  2. Bustinice H (2000) Indicator of inclusion grade for interval-valued fuzzy sets. Application for approximate reasoning based on interval-valued fuzzy sets. Int J Approx Reason 23:137–209MathSciNetCrossRefGoogle Scholar
  3. Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17:191–209CrossRefzbMATHGoogle Scholar
  4. Eļkins A, Šostak A, Uļjane I (2016) On a category of extensional fuzzy rough approximation \(L\)-valued spaces. In: Information processing and management of uncertainty in knowledge-based systems, IPMU2016, Einhofen, The Netherlands, June 20–24, 2016, Proceedings, Part II, pp 48–60Google Scholar
  5. Fang J (2007) I-fuzzy Alexandrov topologies and specialization orders. Fuzzy Sets Syst 158:2359–2374MathSciNetCrossRefzbMATHGoogle Scholar
  6. Goguen JA (1967) \(L\)-fuzzy sets. J Math Anal Appl 18:145–174MathSciNetCrossRefzbMATHGoogle Scholar
  7. Han S-E, Šostak A (2016) \(M\)-valued measure of roughness of \(L\)-fuzzy sets and its topological interpretation. In: Merelo JJ, Rosa A, Cadenas JM, Duarado A, Madani K, Filipe J (eds) Studies in computational intelligence, vol 620. Springer, Berlin, pp 251–266CrossRefGoogle Scholar
  8. Han S-E, Kim IS, Šostak A (2014) On approximate-type systems generated by L-relations. Inf Sci 281:8–20Google Scholar
  9. Hao J, Li Q (2011) The relation between \(L\)-fuzzy rough sets and \(L\)-topology. Fuzzy Sets Syst 178:74–83CrossRefzbMATHGoogle Scholar
  10. Höhle U (1992) \(M\)-valued sets and sheaves over integral commutative CL-monoids. In: Rodabaugh SE, Höhle U, Klement EP (eds) Applications of category theory to fuzzy subsets. Kluwer Academic Publishers, Dordrecht, pp 33–72CrossRefGoogle Scholar
  11. Höhle U (1995) Commutative residuated \(l\)-monoids. In: Höhle U, Klement EP (eds) Nonclassical logics and their applications to fuzzy subsets. Kluwer Academic Publishers, Docrecht, Boston, pp 53–106CrossRefGoogle Scholar
  12. Höhle U (1998) Many-valued equalities, singletons and fuzzy partitions. Soft Comput 2:134–140CrossRefGoogle Scholar
  13. Järvinen J (2002) On the structure of rough approximations. Fundam Inform 53:135–153MathSciNetzbMATHGoogle Scholar
  14. Järvinen J, Kortelainen J (2007) A unified study between modal-like operators, topologies and fuzzy sets. Fuzzy Sets Syst 158:1217–1225CrossRefzbMATHGoogle Scholar
  15. Kehagias A, Konstantinidou M (2003) \(L\)-valued inclusion measure, \(L\)-fuzzy similarity, and \(L\)-fuzzy distance. Fuzzy Sets Syst 136:313–332 L-fuzzy set; upper M-rough approximation operator; lower M-rough approximation operator; measure of inclusion; measure of M-rough approximation of an L-fuzzy set; ditopology, LM-ditopologyMathSciNetCrossRefzbMATHGoogle Scholar
  16. Klawonn F (2000) Fuzzy points, fuzzy relations and fuzzy functions. In: Novák V, Perfilieva I (eds) Discovering the world with fuzzy logic. Springer, Berlin, pp 431–453Google Scholar
  17. Klement EP, Mesiar R, Pap E (2000) Triangular norms. Kluwer Academic Publishers, DordrechtCrossRefzbMATHGoogle Scholar
  18. Kortelainen J (1994) On relationship between modified sets, topological spaces and rough sets. Fuzzy Sets Syst 61:91–95MathSciNetCrossRefzbMATHGoogle Scholar
  19. Lai H, Zhang D (2006) Fuzzy preoder and fuzzy topology. Fuzzy Sets Syst 157:1865–1885CrossRefzbMATHGoogle Scholar
  20. Menger K (1951) Probabilistic geometry. Proc NAS 27:226–229MathSciNetCrossRefzbMATHGoogle Scholar
  21. Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341–356CrossRefzbMATHGoogle Scholar
  22. Pu P, Liu Y (1980) Fuzzy topology I: neighborhood structure of a fuzzy point. J Math Anal Appl 76:571–599MathSciNetCrossRefzbMATHGoogle Scholar
  23. Qin K, Pei Z (2005) On the topological properties of fuzzy rough sets. Fuzzy Sets Syst 151:601–613MathSciNetCrossRefzbMATHGoogle Scholar
  24. Radzikowska AM, Kerre EE (2002) A comparative study of fuzzy rough sets. Fuzzy Sets Syst 126:137–155MathSciNetCrossRefzbMATHGoogle Scholar
  25. Rosenthal KI (1990) Quantales and their applications, Pitman research notes in mathematics, vol 234. Longman Scientific and Technical, HarlowGoogle Scholar
  26. Schweitzer B, Sklar A (1983) Probabilistic metric spaces. North Holland, New YorkzbMATHGoogle Scholar
  27. Skowron A (1988) On the topology in information systems. Bull Pol Acad Sci Math 36:477–480MathSciNetzbMATHGoogle Scholar
  28. Šostak A (1985) On a fuzzy topological structure. Suppl Rend Circ Matem Palermo Ser II 11:125–186zbMATHGoogle Scholar
  29. Šostak A (2010) Towards the theory of M-approximate systems: fundamentals and examples. Fuzzy Sets Syst 161:2440–2461MathSciNetCrossRefzbMATHGoogle Scholar
  30. Tiwari SP, Srivastava AK (2013) Fuzzy rough sets. Fuzzy preoders and fuzzy topoloiges. Fuzzy Sets Syst 210:63–68CrossRefzbMATHGoogle Scholar
  31. Valverde L (1985) On the structure of F-indistinguishability operators. Fuzzy Sets Syst 17:313–328MathSciNetCrossRefzbMATHGoogle Scholar
  32. Wiweger A (1988) On topological rough sets. Bull Pol Acad Sci Math 37:51–62MathSciNetGoogle Scholar
  33. Yao YY (1998a) A comparative study of fuzzy sets and rough sets. Inf Sci 109:227–242MathSciNetCrossRefzbMATHGoogle Scholar
  34. Yao YY (1998b) On generalizing Pawlak approximation operators. In: Proceedings of the first international conference on rough sets and current trends in computing, pp 298–307Google Scholar
  35. Zadeh L (1971) Similarity relations and fuzzy orderings. Inf Sci 3:177–200MathSciNetCrossRefzbMATHGoogle Scholar
  36. Zadeh (1965) Fuzzy sets, information and control 8:338–353Google Scholar
  37. Zeng W, Li H (2006) Inclusion measures, similarity measures and the fuzziness of fuzzy sets and their relations. Int J Intell Syst 21:639–653CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Mathematics Education, Institute of Pure and Applied MathematicsChonbuk National UniversityJeonjuRepublic of Korea
  2. 2.Institute of Mathematics and CSUniversity of LatviaRigaLatvia
  3. 3.Faculty of Physics and MathematicsUniversity of LatviaRigaLatvia

Personalised recommendations