Some properties of B -operations

Bohdan Butkiewicz

Kybernetika (2007)

  • Volume: 43, Issue: 2, page 197-208
  • ISSN: 0023-5954

Abstract

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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.

How to cite

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Butkiewicz, Bohdan. "Some properties of $B$-operations." Kybernetika 43.2 (2007): 197-208. <http://eudml.org/doc/33851>.

@article{Butkiewicz2007,
abstract = {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.},
author = {Butkiewicz, Bohdan},
journal = {Kybernetika},
keywords = {fuzzy system; connectives; weak operations; triangular norms; fuzzy system; connectives; weak operations; triangular norms},
language = {eng},
number = {2},
pages = {197-208},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Some properties of $B$-operations},
url = {http://eudml.org/doc/33851},
volume = {43},
year = {2007},
}

TY - JOUR
AU - Butkiewicz, Bohdan
TI - Some properties of $B$-operations
JO - Kybernetika
PY - 2007
PB - Institute of Information Theory and Automation AS CR
VL - 43
IS - 2
SP - 197
EP - 208
AB - 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.
LA - eng
KW - fuzzy system; connectives; weak operations; triangular norms; fuzzy system; connectives; weak operations; triangular norms
UR - http://eudml.org/doc/33851
ER -

References

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  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 (1998) 
  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 MR2019194
  4. Butkiewicz B. S., Fuzzy systems with weak basic operations, In: Proc. East-West Fuzzy Colloquium 2005 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 Zbl1032.03043MR2015161
  7. Klement E. P., Mesiar, R., Pap E., Triangular Norms, Kluwer Academic Publishers, Dordrecht 2000 Zbl1087.20041MR1790096
  8. Mesiar R., Reusch, B., Thielle H., Fuzzy equivalence relations and fuzzy partitions, J. Multiple-Valued Logic & Soft Computing 12 (2006), 167–181 MR2211529
  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 (1988) MR0931863

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