Distributivity of strong implications over conjunctive and disjunctive uninorms

Daniel Ruiz-Aguilera; Joan Torrens

Kybernetika (2006)

  • Volume: 42, Issue: 3, page 319-336
  • ISSN: 0023-5954

Abstract

top
This paper deals with implications defined from disjunctive uninorms U by the expression I ( x , y ) = U ( N ( x ) , y ) where N is a strong negation. The main goal is to solve the functional equation derived from the distributivity condition of these implications over conjunctive and disjunctive uninorms. Special cases are considered when the conjunctive and disjunctive uninorm are a t -norm or a t -conorm respectively. The obtained results show a lot of new solutions generalyzing those obtained in previous works when the implications are derived from t -conorms.

How to cite

top

Ruiz-Aguilera, Daniel, and Torrens, Joan. "Distributivity of strong implications over conjunctive and disjunctive uninorms." Kybernetika 42.3 (2006): 319-336. <http://eudml.org/doc/33808>.

@article{Ruiz2006,
abstract = {This paper deals with implications defined from disjunctive uninorms $U$ by the expression $I(x,y)=U(N(x),y)$ where $N$ is a strong negation. The main goal is to solve the functional equation derived from the distributivity condition of these implications over conjunctive and disjunctive uninorms. Special cases are considered when the conjunctive and disjunctive uninorm are a $t$-norm or a $t$-conorm respectively. The obtained results show a lot of new solutions generalyzing those obtained in previous works when the implications are derived from $t$-conorms.},
author = {Ruiz-Aguilera, Daniel, Torrens, Joan},
journal = {Kybernetika},
keywords = {$t$-norm; $t$-conorm; uninorm; implication operator; S-implication; R-implication; distributivity; t-norm; t-conorm; uninorm; -implication; -implication; distributivity},
language = {eng},
number = {3},
pages = {319-336},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Distributivity of strong implications over conjunctive and disjunctive uninorms},
url = {http://eudml.org/doc/33808},
volume = {42},
year = {2006},
}

TY - JOUR
AU - Ruiz-Aguilera, Daniel
AU - Torrens, Joan
TI - Distributivity of strong implications over conjunctive and disjunctive uninorms
JO - Kybernetika
PY - 2006
PB - Institute of Information Theory and Automation AS CR
VL - 42
IS - 3
SP - 319
EP - 336
AB - This paper deals with implications defined from disjunctive uninorms $U$ by the expression $I(x,y)=U(N(x),y)$ where $N$ is a strong negation. The main goal is to solve the functional equation derived from the distributivity condition of these implications over conjunctive and disjunctive uninorms. Special cases are considered when the conjunctive and disjunctive uninorm are a $t$-norm or a $t$-conorm respectively. The obtained results show a lot of new solutions generalyzing those obtained in previous works when the implications are derived from $t$-conorms.
LA - eng
KW - $t$-norm; $t$-conorm; uninorm; implication operator; S-implication; R-implication; distributivity; t-norm; t-conorm; uninorm; -implication; -implication; distributivity
UR - http://eudml.org/doc/33808
ER -

References

top
  1. Balasubramaniam J., Rao C. J. M., 10.1109/TFUZZ.2004.825075, IEEE Trans. Fuzzy Systems 12 (2004), 194–198 DOI10.1109/TFUZZ.2004.825075
  2. Combs W. E., 10.1109/91.660804, IEEE Trans. Fuzzy Systems 6 (1998), 1–11 (1998) DOI10.1109/91.660804
  3. Combs W. E., Andrews J. E., 10.1109/TFUZZ.1999.771094, IEEE Trans. Fuzzy Systems 7 (1999), 371 (1999) DOI10.1109/TFUZZ.1999.771094
  4. Combs W. E., Andrews J. E., 10.1109/TFUZZ.1999.771094, IEEE Trans. Fuzzy Systems 7 (1999), 478–479 (1999) DOI10.1109/TFUZZ.1999.771094
  5. Baets B. De, Fodor J. C., 10.1007/s005000050057, Soft Computing 3 (1999), 89–100 (1999) DOI10.1007/s005000050057
  6. Dick S., Kandel A., 10.1109/91.784213, IEEE Trans. Fuzzy Systems 7 (1999), 475–477 (1999) DOI10.1109/91.784213
  7. Fodor J. C., Yager R. R., Rybalov A., 10.1142/S0218488597000312, Internat. J. Uncertainty, Fuzziness and Knowledge-based Systems 5 (1997), 4, 411–427 (1997) Zbl1232.03015MR1471619DOI10.1142/S0218488597000312
  8. González M., Ruiz, D., Torrens J., Algebraic properties of fuzzy morphological operators based on uninorms, In: Artificial Intelligence Research and Development (I. Aguiló, L. Valverde, and M. Escrig, eds.), IOS Press 2003, pp. 27–38 
  9. Klement E. P., Mesiar, R., Pap E., Triangular Norms, Kluwer Academic Publishers, Dordrecht 2000 Zbl1087.20041MR1790096
  10. Martín J., Mayor, G., Torrens J., 10.1016/S0165-0114(02)00430-X, Fuzzy Sets and Systems 137 (2003), 1, 27–42 Zbl1022.03038MR1992696DOI10.1016/S0165-0114(02)00430-X
  11. Mas M., Mayor, G., Torrens J., The distributivity condition for uninorms and t -operators, Fuzzy Sets and Systems 128 (2002), 209–225 Zbl1005.03047MR1908427
  12. Mas M., Mayor, G., Torrens J., Corrigendum to “The distributivity condition for uninorms and t -operators”, Fuzzy Sets and Systems 128 (2002), 209–225, Fuzzy Sets and Systems 153 (2005), 297–299 MR1908427
  13. Mendel J. M., Liang Q., 10.1109/91.771093, IEEE Trans. Fuzzy Systems 7 (1999), 369–371 (1999) DOI10.1109/91.771093
  14. Ruiz D., Torrens J., 10.1142/S0218488503002168, Internat. J. Uncertainty, Fuzziness and Knowledge-Based Systems 11 (2003), 413–428 Zbl1074.03026MR2007849DOI10.1142/S0218488503002168
  15. Ruiz D., Torrens J., Residual implications and co-implications from idempotent uninorms, Kybernetika 40 (2004), 21–38 MR2068596
  16. Ruiz D., Torrens J., 10.1109/TFUZZ.2005.864087, IEEE Trans. Fuzzy Systems 14 (2006), 180–190 DOI10.1109/TFUZZ.2005.864087
  17. Ruiz D., Torrens J., Distributive residual implications from uninorms, In: Proc. EUSFLAT-2005, Barcelona 2005, pp. 369–374 
  18. Trillas E., Alsina C., On the law [ p ∧ q → r ] ≡ [ ( p → r ) ∨ ( q → r ) ] in fuzzy logic, IEEE Trans. Fuzzy Systems 10 (2002), 84–88 

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.