Soft Computing

, Volume 21, Issue 24, pp 7269–7284 | Cite as

Fuzzy objects in spaces with fuzzy partitions

  • Jiří Močkoř
  • Michal Holčapek


A theory of fuzzy objects is derived in the category SpaceFP of spaces with fuzzy partitions, which generalize classical fuzzy sets and extensional maps in sets with similarity relations. It is proved that fuzzy objects in SpaceFP can be characterized by some morphisms in the category of sets with similarity relations. A powerset object functor \({\mathcal {F}}\) in the category SpaceFP is introduced and it is proved that \({\mathcal {F}}\) defines a CSLAT-powerset theory in the sense of Rodabaugh.


Space with a fuzzy partition Category of spaces with fuzzy partitions Fuzzy objects in sets with fuzzy partitions Powerset theory 



This study was funded by the Centre of Excellence Project LQ1602.

Compliance with ethical standards

Conflict of interest

Author declares that he has no conflict of interest.

Human and animal rights

This article does not contain any studies with human participants or animals performed by the authors.


  1. Di Martino F et al (2008) An image coding/decoding method based on direct and inverse fuzzy tranforms. Int J Approx Reason 48:110–131CrossRefzbMATHGoogle Scholar
  2. Garmendia L et al (2009) An algorithm to compute the transitive closure, a transitive approximation and a transitive opening of a fuzzy proximity. Mathw Soft Comput 16:175–191zbMATHMathSciNetGoogle Scholar
  3. Gerla G, Scarpati L (1998) Extension principles for fuzzy set theory. J Inf Sci 106:49–69CrossRefzbMATHMathSciNetGoogle Scholar
  4. Höhle U (2007) Fuzzy sets and sheaves. Part I, basic concepts. Fuzzy Sets Syst 158:1143–1174CrossRefzbMATHGoogle Scholar
  5. Khastan A, Perfilieva I, Alijani Z (2016) A new fuzzy approximation method to Cauchy problem by fuzzy transform. Fuzzy Sets Syst 288:75–95Google Scholar
  6. Močkoř J (2012) Fuzzy sets and cut systems in a category of sets with similarity relations. Soft Comput 16:101–107CrossRefzbMATHGoogle Scholar
  7. Močkoř J (2016) Powerset operators of fuzzy objects. Czech Math J (to appear)Google Scholar
  8. Močkoř J Spaces with fuzzy partitions and fuzzy transform (to appear)Google Scholar
  9. Nguyen HT (1978) A note on the extension principle for fuzzy sets. J Math Anal Appl 64:369–380Google Scholar
  10. Novák V, Perfilijeva I, Močkoř J (1999) Mathematical principles of fuzzy logic. Kluwer Academic Publishers, BostonCrossRefzbMATHGoogle Scholar
  11. Perfilieva I (2006a) Fuzzy transforms and their applications to image compression, Lecture notes in computer science, pp 19–31Google Scholar
  12. Perfilieva I (2006b) Fuzzy transforms: theory and applications. Fuzzy Sets Syst 157:993–1023CrossRefzbMATHMathSciNetGoogle Scholar
  13. Perfilieva I, Novak V, Dvořak A (2008) Fuzzy transforms in the analysis of data. Int J Approx Reason 48:36–46CrossRefzbMATHGoogle Scholar
  14. Perfilieva I, Singh AP, Tiwari SP (2015) On the relationship among F-transform, fuzzy rough set and fuzzy topology. In: Proceedings of IFSA-EUSFLAT. Atlantis Press, Amsterdam, pp 1324–1330Google Scholar
  15. Rodabaugh SE (1997) Powerset operator based foundation for point-set lattice theoretic (poslat) fuzzy set theories and topologies. Quaestiones Mathematicae 20(3):463–530CrossRefzbMATHMathSciNetGoogle Scholar
  16. Rodabaugh SE (2007) Relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice-valued mathematics. Int J Math Math Sci 3:1–71CrossRefzbMATHGoogle Scholar
  17. Rosenthal KI (1990) Quantales and their applications. Pittman Res. Notes in Math. vol 234. Longman, Burnt Mill, HarlowGoogle Scholar
  18. Solovyov SA (2011) Powerset operator foundations for catalc fuzzy set theories. Iran J Fuzzy Syst 8(2):1–46zbMATHMathSciNetGoogle Scholar
  19. Štěpnička M, Valašek R, Numerical solution of partial differential equations with the help of fuzzy transform. In: Proceedings of the FUZZ-IEEE 2005, Reno, Necada, pp 1104–1009Google Scholar
  20. Tiwari SP, Singh Anupam K (2013) Fuzzy preorder fuzzy topology and fuzzy transition system. Logic and Its Applications, Lecture Notes in Computer Science, vol 7750, pp 210–219Google Scholar
  21. Wyler O (1995) Fuzzy logic and categories of fuzzy sets. In: Non-classical logics and their applications to fuzzy subsets. Theory and Decision Library, Series B 32. Kluwer Academic Publishers, Dordrecht, pp 235–268Google Scholar
  22. Yager RR (1996) A characterization of the extension principle. Fuzzy Sets Syst 18:205–217CrossRefzbMATHMathSciNetGoogle Scholar
  23. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institute for Research and Applications of Fuzzy ModelingUniversity of OstravaOstrava 1Czech Republic

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