Geodesic
S-implications and $R$-implications on a finite chain
Kybernetika, Tome 40 (2004) no. 1, pp. 3-20 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

This paper is devoted to the study of two kinds of implications on a finite chain $L$: $S$-implications and $R$-implications. A characterization of each kind of these operators is given and a lot of different implications on $L$ are obtained, not only from smooth t-norms but also from non smooth ones. Some additional properties on these implications are studied specially in the smooth case. Finally, a class of non smooth t-norms including the nilpotent minimum is characterized. Any t-norm in this class satisfies that both, its $S$-implication and its $R$-implication, agree.
This paper is devoted to the study of two kinds of implications on a finite chain $L$: $S$-implications and $R$-implications. A characterization of each kind of these operators is given and a lot of different implications on $L$ are obtained, not only from smooth t-norms but also from non smooth ones. Some additional properties on these implications are studied specially in the smooth case. Finally, a class of non smooth t-norms including the nilpotent minimum is characterized. Any t-norm in this class satisfies that both, its $S$-implication and its $R$-implication, agree.
Classification : 03B52, 06F05, 94D05
Keywords: t-norm; T-conorm; finite chain; smoothness; implication operator 
@article{KYB_2004_40_1_a1,
     author = {Mas, Margarita and Monserrat, Miquel and Torrens, Joan},
     title = {S-implications and $R$-implications on a finite chain},
     journal = {Kybernetika},
     pages = {3--20},
     year = {2004},
     volume = {40},
     number = {1},
     mrnumber = {2068595},
     zbl = {1249.94094},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2004_40_1_a1/}
}
TY  - JOUR
AU  - Mas, Margarita
AU  - Monserrat, Miquel
AU  - Torrens, Joan
TI  - S-implications and $R$-implications on a finite chain
JO  - Kybernetika
PY  - 2004
SP  - 3
EP  - 20
VL  - 40
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/KYB_2004_40_1_a1/
LA  - en
ID  - KYB_2004_40_1_a1
ER  - 
%0 Journal Article
%A Mas, Margarita
%A Monserrat, Miquel
%A Torrens, Joan
%T S-implications and $R$-implications on a finite chain
%J Kybernetika
%D 2004
%P 3-20
%V 40
%N 1
%U http://geodesic.mathdoc.fr/item/KYB_2004_40_1_a1/
%G en
%F KYB_2004_40_1_a1
Mas, Margarita; Monserrat, Miquel; Torrens, Joan. S-implications and $R$-implications on a finite chain. Kybernetika, Tome 40 (2004) no. 1, pp. 3-20. http://geodesic.mathdoc.fr/item/KYB_2004_40_1_a1/

[1] Baets B. De: Model implicators and their characterization. In: Proc. First ICSC Internat. Symposium on Fuzzy Logic, Zürich, Switzerland (N. Steele, ed.). ICSC Academic Press 1995, pp. A42–A49

[2] Baets B. De, Fodor J. C.: On the structure of uninorms and their residual implicators. In: Proc. 18th Linz Seminar on Fuzzy Set Theory, Linz, Austria 1997, pp. 81–87

[3] Baets B. De, Fodor J. C.: Residual operators of representable uninorms. In: Proc. Fifth European Congress on Intelligent Techniques and Soft Computing, Volume 1 (H. J. Zimmermann, ed.), ELITE, Aachen, Germany 1997, pp. 52–56

[4] Baets B. De, Mesiar R.: Residual implicators of continuous t-norms. In: Proc. Fourth European Congress on Intelligent Techniques and Soft Computing, Volume 1 (H. J. Zimmermann, ed.), ELITE, Aachen, Germany 1997, pp. 27–31

[5] Baets B. De, Mesiar R.: Triangular norms on product lattices. Fuzzy Sets and Systems 104 (1999), 61–75 | DOI | MR | Zbl

[6] Bustince H., Burillo, P., Soria F.: Automorphisms, negations and implication operators. Fuzzy Sets and Systems 134 (2003), 209–229 | DOI | MR

[7] Cignoli R., Esteva F., Godo, L., Montagna F.: On a class of left-continuous t-norms. Fuzzy Sets and Systems 131 (2002), 283–296 | DOI | MR

[8] Cignoli R., Esteva F., Godo, L., Torrens A.: Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Computing 4 (2000), 106–112 | DOI

[9] Cignoli R., D’Ottaviano, I., Mundici D.: Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht 2000 | MR | Zbl

[10] Esteva F., Godo L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124 (2001), 271–288 | MR | Zbl

[11] Fodor J. C.: On fuzzy implication operators. Fuzzy Sets and Systems 42 (1991), 293–300 | DOI | MR | Zbl

[12] Fodor J. C.: Contrapositive symmetry of fuzzy implications. Fuzzy Sets and Systems 69 (1995), 141–156 | DOI | MR | Zbl

[13] Fodor J. C.: Smooth associative operations on finite ordinal scales. IEEE Trans. Fuzzy Systems 8 (2000), 791–795 | DOI

[14] Godo L., Sierra C.: A new approach to connective generation in the framework of expert systems using fuzzy logic. In: Proc. XVIIIth ISMVL, Palma de Mallorca, Spain 1988, pp. 157–162

[15] Hájek P.: Mathematics of Fuzzy Logic. Kluwer, Dordrecht 1998

[16] Hájek P.: Basic fuzzy logic and BL-algebras. Soft Computing 2 (1998), 124–128 | DOI

[17] Jenei S.: New family of triangular norms via contrapositive symmetrization of residuated implications. Fuzzy Sets and Systems 110 (2000), 157–174 | MR | Zbl

[18] Mas M., Mayor, G., Torrens J.: $t$-operators and uninorms on a finite totally ordered set. Internat. J. Intelligent Systems (Special Issue: The Mathematics of Fuzzy Sets) 14 (1999), No. 9, 909–922 | DOI | MR | Zbl

[19] Mas M., Monserrat, M., Torrens J.: On left and right uninorms on a finite chain. Fuzzy Sets and Systems, forthcoming | MR | Zbl

[20] Mas M., Monserrat, M., Torrens J.: Operadores de implicación en una cadena finita. In: Proc. Estylf-2002, León, Spain 2002, pp. 297–302

[21] Mas M., Monserrat, M., Torrens J.: On some types of implications on a finite chain. In: Proc. Summer School on Aggregation Operators 2003 (AGOP’2003), Alcalá de Henares, Spain 2003, pp. 107–112

[22] Mayor G., Torrens J.: On a class of operators for expert systems. Internat. J. Intelligent Systems 8 (1993), No. 7, 771–778 | DOI | Zbl

[23] Pei D.: $R_0$ implication: characteristics and applications. Fuzzy Sets and Systems 131 (2002), 297–302 | MR | Zbl

[24] Trillas E., Campo, C. del, Cubillo S.: When QM-operators are implication functions and conditional fuzzy relations. Internat. J. Intelligent Systems 15 (2000), 647–655 | DOI | Zbl