Advertisement

From Lattice Valued Theories to Lattice Valued Analysis

  • Irina PerfilievaEmail author
  • Alexandr Šostak
Chapter
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 325)

Abstract

We claim and justify that the future of a fuzzy logic is in the interconnection of various well-developed theories. We are focused on a lattice valued analysis that unifies the treatments of atomic elements, sets of atomic elements, functions between sets of atomic elements and their properties. We clarify the relationship between a fuzzy function and its ordinary core. We discuss the property of continuity of a fuzzy function in a lattice valued topology.

Keywords

Fuzzy Logic Residuated Lattice Fuzzy Relation Fuzzy Function Extension Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

The support of the two projects is kindly announced: ESF project 2013/0024/1DP/1.1.1.2.0/13/APIA/VIAA/045, European Regional Development Fund in the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070).

References

  1. 1.
    De Baets, B., Mesiar, R.: T-partitions. Fuzzy Sets Syst. 97, 211–223 (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Demirci, M.: Fuzzy functions and their fundamental properties. Fuzzy Sets Syst. 106, 239–246 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Demirci, M.: Fundamentals of m-vague algebra and m-vague arithmetic operations. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 10, 25–75 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)CrossRefzbMATHGoogle Scholar
  5. 5.
    Höhle, U.: Many-valued equalities, singletons and fuzzy partitions. Soft Comput. 2, 134–140 (1998)CrossRefGoogle Scholar
  6. 6.
    Höhle, U., Porst, H.-E., Šostak, A.: Fuzzy functions: a fuzzy extension of the category SET and some related categories. Appl. Gen. Topol. 1, 115–127 (2000)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Klawonn, F.: Fuzzy points, fuzzy relations and fuzzy functions. In: Novák, V., Perfilieva, I. (eds.) Discovering the World with Fuzzy Logic, pp. 431–453. Springer, Berlin (2000)Google Scholar
  8. 8.
    Klawonn, F., Castro, J.L.: Similarity in fuzzy reasoning. Soft Comput Mathw. 2, 197–228 (1995)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Lowen, R.: Fuzzy topological spaces and fuzzy compactness. J. Math. Anal. Appl. 56, 623–633 (1976)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Novák, V.: Fuzzy Sets and Their Applications. Adam Hilger, Bristol (1989)zbMATHGoogle Scholar
  11. 11.
    Perfilieva, I.: Fuzzy function as an approximate solution to a system of fuzzy relation equations. Fuzzy Sets Syst. 147, 363–383 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Perfilieva, I.: Fuzzy function: theoretical and practical point of view. In: Proceedings of 7th Conference European Society for Fuzzy Logic and Technology, EUSFLAT 2011, Aix-Les-Bains, France, 18–22 July 2011, pp. 480–486, Atlantis Press (2011)Google Scholar
  13. 13.
    Perfilieva, I., Dubois, D., Prade, H., Godo, L., Esteva, F., Hokov, P.: Interpolation of fuzzy data. Analytical approach and overview. Fuzzy Sets Syst. 192, 134–158 (2012)CrossRefzbMATHGoogle Scholar
  14. 14.
    Šostak, A.: Fuzzy functions and an extension of the category L-top of Chang-Goguen L-Topological spaces. In: Proceedings of 9th Prague Topological Symposium, Czech Republic, 19–25 August 2001, pp. 271–294 (2001)Google Scholar
  15. 15.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning I, II, III. Inf. Sci. 8–9, 199–257, 301–357, 43–80 (1975)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute for Research and Applications of Fuzzy Modelling, Centre of Excellence IT4Innovations division of the University of OstravaUniversity of OstravaOstrava 1Czech Republic
  2. 2.Department of Mathematics, Faculty of Physics and MathematicsUniversity of LatviaRigaLatvia

Personalised recommendations