Kybernetika 57 no. 2, 332-351, 2021

Some notes on the category of fuzzy implications on bounded lattices

Amin Yousefi, Mashaallah Mashinchi and Radko MesiarDOI: 10.14736/kyb-2021-2-0332


In this paper, we introduce the product, coproduct, equalizer and coequalizer notions on the category of fuzzy implications on a bounded lattice that results in the existence of the limit, pullback, colimit and pushout. Also isomorphism, monic and epic are introduced in this category. Then a subcategory of this category, called the skeleton, is studied. Where none of any two fuzzy implications are $\Phi$-conjugate.


t-norm, fuzzy implication, category, skeleton of category


03B52, 03E72


  1. J. Adámek, H. Herrlich and G. E. Strecker: Abstract and Concrete Categories: The Joy of Cats. Wiley, 1990.   CrossRef
  2. M. Baczynski: On the applications of fuzzy implication functions. In: Soft Computing Applications, Springer, Berlin Heidelberg 2013, pp. 9-10.   DOI:10.1007/978-3-642-33941-7\_4
  3. M. Baczynski, G. Beliakov, H. B. Sola and A. Pradera: Advances in Fuzzy Implication Functions. Springer, Berlin Heidelberg 2013.   CrossRef
  4. M. Baczynski and B. Jayaram: Fuzzy Implications. Springer, Berlin Heidelberg 2008.   CrossRef
  5. B. C. Bedregal: Bounded lattice t-norms as an interval category. In: International Workshop on Logic, Language, Information, and Computation, Springer 2007, pp. 26-37.   CrossRef
  6. R. Bělohlávek: Granulation and granularity via conceptual structures: A perspective from the point of view of fuzzy concept lattices. In: Data mining, rough sets and granular computing, Springer, Berlin Heidelberg 2002, pp. 265-289.   DOI:10.1007/978-3-7908-1791-1\_13
  7. G. Birkhoff: Lattice Theory. American Mathematical Society, Rhode Island 1940.   CrossRef
  8. B. A. Davey and H. A. Priestley: Introduction to Lattices and Order. Cambridge University Press 2002.   CrossRef
  9. B. De Baets and R. Mesiar: Triangular norms on product lattices Fuzzy Sets and Systems 104 (1999),1, 61-75.   DOI:10.1016/s0165-0114(98)00259-0
  10. B. De Baets and R. Mesiar: Discrete Triangular Norms. In:Topological and Algebraic Structures in Fuzzy Sets, Springer 2003, pp. 389-400.   DOI:10.1007/978-94-017-0231-7\_16
  11. P. Hájek: Metamathematics of Fuzzy Logic. Volume 4, Springer Science and Business Media, 2013.   CrossRef
  12. E P. Klement, R. Mesiar and E. Pap: Triangular Norms Volume 8, Springer Science and Business Media, 2013.   CrossRef
  13. C. C. Lee: Fuzzy logic in control systems: fuzzy logic controller. IEEE Trans. Systems Man Cybernet. 20 (1990), 2, 404-418.   DOI:10.1109/21.52551
  14. G. Mayor, J. Suñer and J. Torrens: Operations on Finite Settings: from Triangular Norms to Copulas. In: Copulas and Dependence Models with Applications, Springer 2017, pp. 157-170.   DOI:10.1007/978-3-319-64221-5\_10
  15. H. T. Nguyen and E. A. Walker: A first Course in Fuzzy Logic. Chapman and Hall/CRC Press 2006.   CrossRef
  16. B. C. Pierce, M. R. Garey and A. Meyer: Basic Category Theory for Computer Scientists. MIT Press 1991.   CrossRef
  17. M. Togai and H. Watanabe: Expert system on a chip: An engine for real-time approximate reasoning. In: Proc. ACM SIGART international symposium on Methodologies for intelligent systems, ACM 1986, pp. 147-154.   CrossRef
  18. A. Yousefi and M. Mashinchi: Categories of fuzzy implications and R-implications on bounded lattices. In: 6th Iranian Joint Congress on Fuzzy and Intelligent Systems (CFIS), IEEE 2018, pp. 40-42.   DOI:10.1109/cfis.2018.8336622
  19. A. Yousefi and M. Mashinchi: Counting T-norms and R-implications on Bounded Lattices. In: 9th National Conference on Mathematics of Payame Noor University, On CD 2019, pp. 726-731.   CrossRef
  20. Y. Yu, J. N. Mordeson and S. C. Cheng: Elements of L-algebra. Lecture Notes in Fuzzy Mathematics and Computer Science, Creighton University, Omaha 1994.   CrossRef
  21. L. A. Zadeh: A computational approach to fuzzy quantifiers in natural languages. Computers Math. Appl. 9, Elsevier (1983), 149-184.   DOI:10.1016/0898-1221(83)90013-5