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Using single axioms to characterize L-rough approximate operators with respect to various types of L-relations

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Abstract

Considering L being a complete Heyting algebra, this paper mainly proposes a general framework of L-rough approximate operators in which constructive and axiomatic approaches are used. In the constructive approach, upper and lower L-rough approximate operators are introduced and their connections with L-relations are investigated. In the axiomatic approach, various types of set-theoretic L-operators are defined. It is shown that each type of L-rough approximate operators corresponding to special kind of L-relations, including serial, reflexive, symmetric, transitive, mediate, Euclidean and adjoint L-relations as well as their compositions, can be characterized by single axioms.

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References

  1. Bělohlávek R (2004) Concept lattics and order in fuzzy logic. Ann Pure Appl Logic 128:277–298

    MathSciNet  MATH  Google Scholar 

  2. Bao YL, Yang HL, She YH (2018) Using one axiom to characterize L-fuzzy rough approximation operators based on residuated lattices. Fuzzy Sets Syst 336:87–115

    MathSciNet  MATH  Google Scholar 

  3. Goguen JA (1967) \(L\)-subsets. J Math Anal Appl 18:145–174

    MathSciNet  MATH  Google Scholar 

  4. Han S-E, Kim IS, Šostak A (2014) On approximate-type systems generated by \(L\)-relations. Inf Sci 281:8–21

    MathSciNet  MATH  Google Scholar 

  5. Han S-E, Šostak A (2018) On the measure of \(M\)-rough approximation of \(L\)-fuzzy sets. Soft Comput 22(12):3843–3855

    MATH  Google Scholar 

  6. Li LQ (2019) p-Topologicalness—a relative topologicalness in \(\top\)-convergence spaces. Mathematics 7(3):228

    Google Scholar 

  7. Li Q-H, Li H-Y (2019) Applications of fuzzy inclusion orders between \(L\)-subsets in fuzzy topological structures. J Intell Fuzzy Syst 37(2):2587–2596

    Google Scholar 

  8. Li Q-H, Huang H-L, Xiu Z-Y (2019) Degrees of special mappings in the theory of \(L\)-convex spaces. J Intell Fuzzy Syst 37(2):2265–2274

    Google Scholar 

  9. Lin TY (1992) Topological and fuzzy rough sets. In: Slowinski R (ed) Decision support by experience-application of the rough set theory. Kluwer Academic Publishers, Boston, pp 287–304

    Google Scholar 

  10. Lin TY, Liu Q (1994) Rough approximate operators: axiomatic rough set theory. In: Ziarko W (ed) Rough sets fuzzy sets and knowledge discovery. Springer, Berlin, pp 256–260

    Google Scholar 

  11. Lin TY (1996) A rough logic formalism for fuzzy controllers: a hard and soft computing view. Int J Approx Reason 15:395–414

    MATH  Google Scholar 

  12. Liu GL (2006) The axiomatization of the rough set upper approximation operations. Fundam Inform 69(23):331–342

    MathSciNet  MATH  Google Scholar 

  13. Liu GL (2008) Generalized rough sets over fuzzy lattices. Inf Sci 178(6):1651–1662

    MathSciNet  MATH  Google Scholar 

  14. Liu GL, Sai Y (2010) Invertible approximation operators of generalized rough sets and fuzzy rough sets. Inf Sci 180:2221–2229

    MathSciNet  MATH  Google Scholar 

  15. Liu GL (2013) Using one axiom to characterize rough set and fuzzy rough set approximations. Inf Sci 223:285–296

    MathSciNet  MATH  Google Scholar 

  16. Mi JS, Zhang WX (2004) An axiomatic characterization of a fuzzy generalization of rough sets. Inf Sci 160:235–249

    MathSciNet  MATH  Google Scholar 

  17. Mi JS, Leung Y, Wu WZ (2005) An uncertainty measure in partition-based fuzzy rough sets. Int J Gen Syst 34:77–90

    MathSciNet  MATH  Google Scholar 

  18. Mi JS, Leung Y, Zhao HY, Feng T (2008) Generalized fuzzy rough sets determined by a triangular norm. Inf Sci 178:3203–3213

    MathSciNet  MATH  Google Scholar 

  19. Morsi NN, Yakout MM (1998) Axiomatics for fuzzy rough sets. Fuzzy Sets Syst 100:327–342

    MathSciNet  MATH  Google Scholar 

  20. Nguyen HT (1992) Intervals in Boolean rings: approximation and logic. Found Comput Decis Sci 17:131–138

    MathSciNet  MATH  Google Scholar 

  21. Pang B (2018) Categorical properties of \(L\)-fuzzifying convergence spaces. Filomat 32(11):4021–4036

    MathSciNet  Google Scholar 

  22. Pang B (2019) Convergence structures in \(M\)-fuzzifying convex spaces. Quaest Math. https://doi.org/10.2989/16073606.2019.1637379

    Article  Google Scholar 

  23. Pang B, Mi J-S, Xiu Z-Y (2019) \(L\)-fuzzifying approximation operators in fuzzy rough sets. Inf Sci 480:14–33

    MathSciNet  Google Scholar 

  24. Pang B, Mi J-S, Yao W (2019) \(L\)-fuzzy rough approximation operators via three types of \(L\)-fuzzy relation. Soft Comput 23:11433–11446

    Google Scholar 

  25. Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341–356

    MATH  Google Scholar 

  26. Radzikowska AM, Kerre EE (2002) A comparative study of fuzzy rough sets. Fuzzy Sets Syst 126:137–155

    MathSciNet  MATH  Google Scholar 

  27. Radzikowska AM, Kerre EE (2004) Fuzzy rough sets based on residuated lattices. Trans Rough Sets II LNCS 3135:278–296

    MATH  Google Scholar 

  28. Radzikowska AM (2010) On lattice-based fuzzy rough sets, 35 years of fuzzy set theory. Springer, Berlin Heidelberg, pp 107–126

    MATH  Google Scholar 

  29. She Y, Wang G (2009) An axiomatic approach of fuzzy rough sets based on residuated lattices. Comput Math Appl 58:189–201

    MathSciNet  MATH  Google Scholar 

  30. She Y, He X (2014) Rough approximation operators on \(R_0\)-algebras (nilpotent minimum algebras) with an application in formal logic \({\cal{L}}^*\). Inf Sci 277:71–89

    MATH  Google Scholar 

  31. Slowinski R, Vanderpooten D (2000) A generalized definition of rough approximations based on similarity. IEEE Trans Knowl Data Eng 12(2):331–336

    Google Scholar 

  32. Šostak A (2010) Towards the theory of \(\mathbb{M}\)-approximate systems: fundamentals and examples. Fuzzy Sets Syst 161:2440–2461

    MathSciNet  MATH  Google Scholar 

  33. Thiele H (2000) On axiomatic characterizations of crisp approximation operators. Inf Sci 129:221–226

    MATH  Google Scholar 

  34. Thiele H (2000) On axiomatic characterization of fuzzy approximation operators. I, the fuzzy rough set based case, RSCTC2000, Banff Park Lodge, Bariff, Canada, October 19, In: Conf. Proc., pp 239–247

  35. Thiele H (2001) On axiomatic characterization of fuzzy approximation operators II, the rough fuzzy set based case. In: Proc. 31st IEEE Internat. Symp. on multiple-valued logic, pp 330–335

  36. Wang B, Li Q, Xiu Z-Y (2019) A categorical approach to abstract convex spaces and interval spaces. Open Math 17:374–384

    MathSciNet  MATH  Google Scholar 

  37. Wang CY (2018) Single axioms for lower fuzzy rough approximation operators determined by fuzzy implications. Fuzzy Sets Syst 336:116–147

    MathSciNet  MATH  Google Scholar 

  38. Wang L, Wu X-Y, Xiu Z-Y (2019) A degree approach to relationship among fuzzy convex structures, fuzzy closure systems and fuzzy Alexandrov topologies. Open Math 17:913–928

    MathSciNet  MATH  Google Scholar 

  39. Wu WZ, Mi JS, Zhang WX (2003) Generalized fuzzy rough sets. Inf Sci 151:263–282

    MathSciNet  MATH  Google Scholar 

  40. Wu WZ, Zhang WX (2004) Constructive and axiomatic approaches of fuzzy approximation operators. Inf Sci 159:233–254

    MathSciNet  MATH  Google Scholar 

  41. Wu WZ, Leung Y, Mi JS (2005) On characterizations of \(({\mathscr {I}}, {\mathscr {T}})\)-fuzzy rough approximation operators. Fuzzy Sets Syst 154:76–102

    MathSciNet  Google Scholar 

  42. Wu WZ, Leung Y, Shao MW (2013) Generalized fuzzy rough approximation operators determined by fuzzy implicators. Int J Approx Reason 54:1388–1409

    MathSciNet  MATH  Google Scholar 

  43. Wu WZ, Li TJ, Gu SM (2015) Using one axiom to characterize fuzzy rough approximation operators determined by a fuzzy implication operator. Fundam Inf 142:87–104

    MathSciNet  MATH  Google Scholar 

  44. Wu WZ, Xu YH, Shao MW, Wang GY (2016) Axiomatic characterizations of \((S, T)\)-fuzzy rough approximation operators. Inf Sci 334–335:17–43

    MATH  Google Scholar 

  45. Wu WZ, Xu MW, Wang X (2019) Using single axioms to characterize \((S, T)\)-intuitionistic fuzzy rough approximation operators. Int J Mach Learn Cybern 10:27–42

    Google Scholar 

  46. Wybraniec-Skardowska U (1989) On a generalization of approximation space. Bull Polish Acad Sci Math 37:51–61

    MathSciNet  MATH  Google Scholar 

  47. Yao W, She Y, Lu L-X (2019) Metric-based \(L\)-fuzzy rough sets: approximation operators and definable sets. Knowl Based Syst 163:91–102

    Google Scholar 

  48. Yao YY (1996) Two views of the theory of rough sets infinite universe. Int J Approx Reason 15:291–317

    MATH  Google Scholar 

  49. Yao YY (1998) Constructive and algebraic methods of the theory of rough sets. Inf Sci 109:21–47

    MathSciNet  MATH  Google Scholar 

  50. Yao YY (1998) Relational interpretations of neighborhood operators and rough set approximation operators. Inf Sci 111:239–259

    MathSciNet  MATH  Google Scholar 

  51. Yao YY, Lin TY (1996) Generalization of rough sets using modal logic. Intell Autom Soft Comput 2(2):103–120

    Google Scholar 

  52. Zhang HY, Song HJ (2017) On simplified axiomatic characterizations of \((\theta, \sigma )\)-fuzzy rough approximation operators. Int J Uncertain Fuzziness Knowl Based Syst 25(3):457–476

    MathSciNet  Google Scholar 

  53. Zhang K, Zhan J, Yao YY (2019) TOPSIS method based on a fuzzy covering approximation space: an application to biological nano-materials selection. Inf Sci 502:297–329

    Google Scholar 

  54. Zhang L, Zhan J, Xu ZX, Alcantud JCR (2019) Covering-based general multigranulation intuitionistic fuzzy rough sets and corresponding applications to multi-attribute group decision-making. Inf Sci 494:114–140

    Google Scholar 

  55. Zhang YL, Luo MK (2011) On minimization of axiom sets characterizing covering-based approximation operators. Inf Sci 181:3032–3042

    MathSciNet  MATH  Google Scholar 

  56. Zhang X, Mei C, Chen D, Li J (2016) Feature selection in mixed data: a method using a novel fuzzy rough set-based information entropy. Pattern Recognit 56:1–15

    MATH  Google Scholar 

  57. Zhu W (2007) Generalized rough sets based on relations. Inf Sci 177:4997–5011

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express their sincere thanks to the Editors and anonymous reviewers for their most valuable comments and suggestions in improving this paper greatly. This work is supported by the Natural Science Foundation of China (Nos. 11701122, 61573127, 61170107) and Beijing Institute of Technology Research Fund Program for Young Scholars (No. 2019CX04111).

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Pang, B., Mi, JS. Using single axioms to characterize L-rough approximate operators with respect to various types of L-relations. Int. J. Mach. Learn. & Cyber. 11, 1061–1082 (2020). https://doi.org/10.1007/s13042-019-01051-z

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