Kybernetika 48 no. 1, 144-164, 2012

On the L-valued categories of L-E-ordered sets

Olga Grigorenko


The aim of this paper is to construct an $L$-valued category whose objects are $L$-$E$-ordered sets. To reach the goal, first, we construct a category whose objects are $L$-$E$-ordered sets and morphisms are order-preserving mappings (in a fuzzy sense). For the morphisms of the category we define the degree to which each morphism is an order-preserving mapping and as a result we obtain an $L$-valued category. Further we investigate the properties of this category, namely, we observe some special objects, special morphisms and special constructions.


category, $L$-valued category, fuzzy order relation


03E72, 18A05, 18B35


  1. J. Adamek, H. Herrlich and G. E. Strecker: Abstract and concrete categories: The joy of cats. Reprints in Theory and Applications of Categories, No. 17 2006.   CrossRef
  2. R. Bělohlávek: Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic/Plenum Press, New York 2002.   CrossRef
  3. U. Bodenhofer: A similarity-based generalization of fuzzy orderings preserving the classical axioms. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 8(5) (2000), 593-610.   CrossRef
  4. U. Bodenhofer: Representations and constructions of similarity-based fuzzy orderings. Fuzzy Sets and Systems 137 (2003), 113-136.   CrossRef
  5. U. Bodenhofer and J. Küng: Fuzzy orderings in flexible query answering systems. Soft Computing 8 (2004), 7, 512-522.   CrossRef
  6. U. Bodenhofer, B. De Baets and J. Fodor: A compendium of fuzzy weak orders: representations and constructions. Fuzzy Sets and Systems 158 (2007), 593-610.   CrossRef
  7. M. Demirci: A theory of vague lattices based on many-valued equivalence relations - I: General representation results. Fuzzy Sets and Systems 151 (2005), 3, 437-472.   CrossRef
  8. J. Fodor and M. Roubens: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht 1994.   CrossRef
  9. J. A. Goguen: L-fuzzy sets. J. Math. Anal. Appl. 18 (1967), 338-353.   CrossRef
  10. O. Grigorenko: Categorical aspects of aggregation of fuzzy relations. In: Abstracts 10th Conference on Fuzzy Set Theory and Applications, 2010, p. 61.   CrossRef
  11. O. Grigorenko: Degree of monotonicity in aggregation process. In: Proc. 2010 IEEE International Conference on Fuzzy Systems, pp. 1080-1087.   CrossRef
  12. H. Herrlich and G. E. Strecker: Category Theory. Second edition. Heldermann Verlag, Berlin 1978.   CrossRef
  13. U. Höhle and N. Blanchard: Partial ordering in L-underdeterminate sets. Inform. Sci. 35 (1985), 133-144.   CrossRef
  14. U. Höhle: Quotients with respect to similarity relations. Fuzzy Sets and Systems 27 (1988), 31-44.   CrossRef
  15. U. Höhle: M-valued sets and sheaves over integral commutative cl-monoids. In: Applications of Category Theory to Fuzzy Subsets (S. E. Rodabaugh et al., eds.), Kluwer Academic Publishers, Dordrecht, Boston, London, pp. 33-72.   CrossRef
  16. E. P. Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer Academic Publishers, The Netherlands 2002.   CrossRef
  17. H. L. Lai and D. X. Zhang: Many-valued complete distributivity. arXiv:math.CT/0603590, 2006.   CrossRef
  18. F. W. Lawvere: Metric spaces, generalized logic, and closed categories. Rend. Sem. Mat. Fis. Milano 43 (1973), 135-166. Also Reprints in Theory and Applications of Categories 1 (2002).   CrossRef
  19. F. W. Lawvere: Taking categories seriously. Revisita Columbiana de Matemáticas XX (1986), 147-178. Also Reprints in Theory and Applications of Categories 8 (2005).   CrossRef
  20. O. Lebedeva (Grigorenko): Fuzzy order relation and fuzzy ordered set category. In: New Dimensions in fuzzy logic and related technologies. Proc. 5th EUSFLAT Conference, Ostrava 2007, pp. 403-407   CrossRef
  21. S. Ovchinnikov: Similarity relations, fuzzy partitions, and fuzzy orderings. Fuzzy Sets and Systems 40 (1991), 1, 107-126.   CrossRef
  22. A. Sostak: Fuzzy categories versus categories of fuzzy structured sets: Elements of the theory of fuzzy categories. Mathematik-Arbeitspapiere, Universitat Bremen 48 (1997), 407-437.   CrossRef
  23. A. Sostak: $L$-valued categories: Generalities and examples related to algebra and topology. In: Categorical Structures and Their Applications (W. Gahler and G. Preuss, eds.), World Scientific 2004, pp. 291-312.   CrossRef
  24. L. A. Zadeh: Similarity relations and fuzzy orderings. Inform. Sci. 3 (1971), 177-200.   CrossRef
  25. L. A. Zadeh: Fuzzy sets. Inform. Control 8 (1965), 338-353.   CrossRef