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

Olga Grigorenko

Kybernetika (2012)

  • Volume: 48, Issue: 1, page 144-164
  • ISSN: 0023-5954

Abstract

top
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.

How to cite

top

Grigorenko, Olga. "On the $L$-valued categories of $L$-$E$-ordered sets." Kybernetika 48.1 (2012): 144-164. <http://eudml.org/doc/246439>.

@article{Grigorenko2012,
abstract = {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.},
author = {Grigorenko, Olga},
journal = {Kybernetika},
keywords = {category; $L$-valued category; fuzzy order relation; aggregation function; commutative -monoid; epimorphism; extensional map; lattice-valued category; lattice-valued equality; lattice-valued partial order; monomorphism; product of objects of a category; residuation; t-norm; fuzzy order relation; aggregation functions},
language = {eng},
number = {1},
pages = {144-164},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On the $L$-valued categories of $L$-$E$-ordered sets},
url = {http://eudml.org/doc/246439},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Grigorenko, Olga
TI - On the $L$-valued categories of $L$-$E$-ordered sets
JO - Kybernetika
PY - 2012
PB - Institute of Information Theory and Automation AS CR
VL - 48
IS - 1
SP - 144
EP - 164
AB - 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.
LA - eng
KW - category; $L$-valued category; fuzzy order relation; aggregation function; commutative -monoid; epimorphism; extensional map; lattice-valued category; lattice-valued equality; lattice-valued partial order; monomorphism; product of objects of a category; residuation; t-norm; fuzzy order relation; aggregation functions
UR - http://eudml.org/doc/246439
ER -

References

top
  1. J. Adamek, H. Herrlich, G. E. Strecker, Abstract and concrete categories: The joy of cats., Reprints in Theory and Applications of Categories, No. 17 2006. (2006) Zbl1113.18001MR2240597
  2. R. Bělohlávek, Fuzzy Relational Systems: Foundations and Principles., Kluwer Academic/Plenum Press, New York 2002. (2002) Zbl1067.03059
  3. U. Bodenhofer, 10.1142/S0218488500000411, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 8(5) (2000), 593-610. (2000) Zbl1113.03333MR1784649DOI10.1142/S0218488500000411
  4. U. Bodenhofer, Representations and constructions of similarity-based fuzzy orderings., Fuzzy Sets and Systems 137 (2003), 113-136. (2003) Zbl1052.91032MR1992702
  5. U. Bodenhofer, J. Küng, 10.1007/s00500-003-0308-9, Soft Computing 8 (2004), 7, 512-522. (2004) Zbl1063.68047MR1784649DOI10.1007/s00500-003-0308-9
  6. U. Bodenhofer, B. De Baets, J. Fodor, A compendium of fuzzy weak orders: representations and constructions., Fuzzy Sets and Systems 158 (2007), 593-610. (2007) Zbl1119.06001MR2302639
  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. (2005) Zbl1067.06006MR2126168
  8. J. Fodor, M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support., Kluwer Academic Publishers, Dordrecht 1994. (1994) Zbl0827.90002
  9. J. A. Goguen, L-fuzzy sets., J. Math. Anal. Appl. 18 (1967), 338-353. (1967) Zbl0145.24404MR0224391
  10. O. Grigorenko, Categorical aspects of aggregation of fuzzy relations., In: Abstracts 10th Conference on Fuzzy Set Theory and Applications, 2010, p. 61. (2010) 
  11. O. Grigorenko, Degree of monotonicity in aggregation process., In: Proc. 2010 IEEE International Conference on Fuzzy Systems, pp. 1080-1087. (2010) 
  12. H. Herrlich, G. E. Strecker, Category Theory., Second edition. Heldermann Verlag, Berlin 1978. (1978) MR2377903
  13. U. Höhle, N. Blanchard, 10.1016/0020-0255(85)90045-3, Inform. Sci. 35 (1985), 133-144. (1985) Zbl0576.06004MR0794764DOI10.1016/0020-0255(85)90045-3
  14. U. Höhle, 10.1016/0165-0114(88)90080-2, Fuzzy Sets and Systems 27 (1988), 31-44. (1988) Zbl0666.18002MR0950448DOI10.1016/0165-0114(88)90080-2
  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. Zbl0766.03037MR1154568
  16. E. P. Klement, R. Mesiar, E. Pap, Triangular Norms., Kluwer Academic Publishers, The Netherlands 2002. (2002) Zbl1012.03033MR1790096
  17. H. L. Lai, D. X. Zhang, Many-valued complete distributivity., arXiv:math.CT/0603590, 2006. (2006) 
  18. F. W. Lawvere, 10.1007/BF02924844, Rend. Sem. Mat. Fis. Milano 43 (1973), 135-166. Also Reprints in Theory and Applications of Categories 1 (2002). (1973) MR0352214DOI10.1007/BF02924844
  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). (1986) Zbl0648.18001MR0948965
  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 (2007) 
  21. S. Ovchinnikov, 10.1016/0165-0114(91)90048-U, Fuzzy Sets and Systems 40 (1991), 1, 107-126. (1991) Zbl0725.04003MR1103658DOI10.1016/0165-0114(91)90048-U
  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. (1997) 
  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. (2004) Zbl1068.18001MR2127008
  24. L. A. Zadeh, 10.1016/S0020-0255(71)80005-1, Inform. Sci. 3 (1971), 177-200. (1971) Zbl0218.02058MR0297650DOI10.1016/S0020-0255(71)80005-1
  25. L. A. Zadeh, 10.1016/S0019-9958(65)90241-X, Inform. Control 8 (1965), 338-353. (1965) Zbl0139.24606MR0219427DOI10.1016/S0019-9958(65)90241-X

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.