Aggregation Through Composition: Unification of Three Principal Fuzzy Theories

  • Irina PerfilievaEmail author
  • Anand P. Singh
  • S. P. Tiwari
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 981)


This paper shows that the theories of fuzzy rough sets, F-transforms and fuzzy automata can be unified in the framework of fuzzy relational structures. Specifically, the key concepts in such theories are represented as lattice-based aggregations in the form of compositions with suitable fuzzy relations. Furthermore, it is shown that the principal parts of morphisms between all considered fuzzy relational structures coincide.


Fuzzy relation Fuzzy rough set F-transform Fuzzy automata 



This work is supported by University of Ostrava grant lRP201824 “Complex topological structures”. The additional support was also provided by the Czech Science Foundation (GAČR) through the project of No. 18-06915S.


  1. 1.
    Ćirić, M., Ignjatović, J.: Fuzziness in automata theory: why? How? Stud. Fuzziness Soft Comput. 298, 109–114 (2013)CrossRefGoogle Scholar
  2. 2.
    Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. Int. J. Gen. Syst. 17, 191–209 (1990)CrossRefGoogle Scholar
  3. 3.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Boston (1998)CrossRefGoogle Scholar
  4. 4.
    Jin, J., Li, Q., Li, Y.: Algebraic properties of \(L\)-fuzzy finite automata. Inf. Sci. 234, 182–202 (2013)Google Scholar
  5. 5.
    Močkoř, J., Holčapek, M.: Fuzzy objects in spaces with fuzzy partitions. Soft Comput. 21, 7269–7284 (2017)CrossRefGoogle Scholar
  6. 6.
    Perfilieva, I., Valasek, R.: Fuzzy transforms in removing noise. Adv. Soft Comput. 2, 221–230 (2005)CrossRefGoogle Scholar
  7. 7.
    Perfilieva, I.: Fuzzy transforms: theory and applications. Fuzzy Sets Syst. 157, 993–1023 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Perfilieva, I.: Fuzzy transforms: a challenge to conventional transforms. Adv. Image Electron Phys. 147, 137–196 (2007)CrossRefGoogle Scholar
  9. 9.
    Perfilieva, I., Singh, A.P., Tiwari, S.P.: On the relationship among \(F\)-transform, fuzzy rough set and fuzzy topology. Soft Comput. 21, 3513–3523 (2017)Google Scholar
  10. 10.
    Perfilieva, I., Holčapek, M., Kreinovich, V.: A new reconstruction from the \(F\)-transform components. Fuzzy Sets Syst. 288, 3–25 (2016)Google Scholar
  11. 11.
    Qiao, J., Hua, B.Q.: A short note on \(L\)-fuzzy approximation spaces and \(L\)-fuzzy pretopological spaces. Fuzzy Sets Syst. 312, 126–134 (2017)Google Scholar
  12. 12.
    Qiu, D.: Automata theory based on quantum logic: some characterizations. Inf. Comput. 190, 179–195 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Qiu, D.: Automata theory based on complete residuated lattice-valued logic(I). Sci. China 44, 419–429 (2001)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Radzikowska, A.M., Kerre, E.E.: Fuzzy rough sets based on residuated lattices. Lecture Notes in Computer Science, vol. 3135, pp. 278–296 (2005)Google Scholar
  15. 15.
    Santos, E.S.: Maximin automata. Inf. Control. 12, 367–377 (1968)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Singh, A.P., Tiwari, S.P.: Lattice \(F\)-transform for functions in two variables. J. Fuzzy Set Valued Anal. 3, 185–195 (2016)Google Scholar
  17. 17.
    Singh, A. P., Tiwari, S. P., Perfilieva, I.: \(F\)-transforms, \(L\)-fuzzy partitions and \(L\)-fuzzy pretopological spaces: an operator oriented view. Fuzzy Sets Syst. (Submitted)Google Scholar
  18. 18.
    Sussner, P.: Lattice fuzzy transforms from the perspective of mathematical morphology. Fuzzy Sets Syst. 288, 115–128 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Irina Perfilieva
    • 1
    Email author
  • Anand P. Singh
    • 1
  • S. P. Tiwari
    • 2
  1. 1.Institute for Research and Applications of Fuzzy ModelingOstrava 1Czech Republic
  2. 2.Indian Institute of Technology (ISM)DhanbadIndia

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