Geodesic
Some properties of $B$-operations
Kybernetika, Tome 43 (2007) no. 2, pp. 197-208 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In the paper the problem of mathematical properties of $B$-operations and weak $W\!B$-operations introduced by the author for interpretation of connectives “and”, “or”, and “also” in fuzzy rules is considered. In previous author’s papers some interesting properties of fuzzy systems with these operations were shown. These operations are weaker than triangular norms used commonly for a fuzzy system described by set of rules of the type if – then. Monotonicity condition, required for triangular norms, is replaced by condition of positivity (negativity), i. e. operations must be only positively (negatively) defined. Weak $B$-operations may not fulfill associativity condition.
In the paper the problem of mathematical properties of $B$-operations and weak $W\!B$-operations introduced by the author for interpretation of connectives “and”, “or”, and “also” in fuzzy rules is considered. In previous author’s papers some interesting properties of fuzzy systems with these operations were shown. These operations are weaker than triangular norms used commonly for a fuzzy system described by set of rules of the type if – then. Monotonicity condition, required for triangular norms, is replaced by condition of positivity (negativity), i. e. operations must be only positively (negatively) defined. Weak $B$-operations may not fulfill associativity condition.
Classification : 03E72, 20M14, 93C42
Keywords: fuzzy system; connectives; weak operations; triangular norms 
@article{KYB_2007_43_2_a6,
     author = {Butkiewicz, Bohdan},
     title = {Some properties of $B$-operations},
     journal = {Kybernetika},
     pages = {197--208},
     year = {2007},
     volume = {43},
     number = {2},
     mrnumber = {2343395},
     zbl = {1132.03348},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2007_43_2_a6/}
}
TY  - JOUR
AU  - Butkiewicz, Bohdan
TI  - Some properties of $B$-operations
JO  - Kybernetika
PY  - 2007
SP  - 197
EP  - 208
VL  - 43
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/KYB_2007_43_2_a6/
LA  - en
ID  - KYB_2007_43_2_a6
ER  - 
%0 Journal Article
%A Butkiewicz, Bohdan
%T Some properties of $B$-operations
%J Kybernetika
%D 2007
%P 197-208
%V 43
%N 2
%U http://geodesic.mathdoc.fr/item/KYB_2007_43_2_a6/
%G en
%F KYB_2007_43_2_a6
Butkiewicz, Bohdan. Some properties of $B$-operations. Kybernetika, Tome 43 (2007) no. 2, pp. 197-208. http://geodesic.mathdoc.fr/item/KYB_2007_43_2_a6/

[1] Butkiewicz B. S.: Steady-state error of a system with fuzzy controller. IEEE Trans. System Man Cybernet., Part B: Cybernetics 28 (1998), 6, 855–860

[2] Butkiewicz B. S.: Fuzzy control system with $B$-operations. In: Advances in Soft Computing, Developments in Soft Computing (R. John and R. Birkenhead, eds.), Physica–Verlag, Heidelberg – New York 2001, pp. 48–55

[3] Butkiewicz B. S.: About robustness of fuzzy logic PD and PID controller under changes of reasoning methods. In: Advances in Computational Intelligence and Learning, Methods and Applications (H.-J. Zimmermann, G. Tselentis, M. van Someren, and G. Dounias, eds.), Kluwer Academic Publishers, Boston – London 2001, pp. 307–318 | MR

[4] Butkiewicz B. S.: Fuzzy systems with weak basic operations. In: Proc. East-West Fuzzy Colloquium 2005 $12^{{\mathrm th}}$ Zittau Fuzzy Colloquium. Hochschule Zittau/Gorlitz 2005, pp. 73–78

[5] Butkiewicz B. S.: About the best reasoning method for fuzzy control. In: 6th World Congress on Intelligent Control and Automation, Dalian, China, Conf. Proc. Vol. 5, pp. 3920–2924, IEEE Catalog No 06EX1358, 2006

[6] Calvo T., Mesiar R.: Aggregation operators: ordering and bounds. Fuzzy Sets and Systems 139 (2003), 685–697 | MR | Zbl

[7] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000 | MR | Zbl

[8] Mesiar R., Reusch, B., Thielle H.: Fuzzy equivalence relations and fuzzy partitions. J. Multiple-Valued Logic & Soft Computing 12 (2006), 167–181 | MR

[9] Mesiar R.: Personal communication on East-West Fuzzy Colloquium, Zittau 200.

[10] Yager R. R.: On ordered weighted operators in multicriteria decisionmaking. IEEE Trans. System Man Cybernet. 18 (1988), 183–190 | MR