On a functional equation connected to the distributivity of fuzzy implications over triangular norms and conorms

Michał Baczyński; Tomasz Szostok; Wanda Niemyska

Kybernetika (2014)

  • Volume: 50, Issue: 5, page 679-695
  • ISSN: 0023-5954

Abstract

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Distributivity of fuzzy implications over different fuzzy logic connectives have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems (see [9, 15] and [4]). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see [5]) where and is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation where are functions defined on some intervals of satisfying additional assumptions. We analyze the cases when is injective and when is not injective.

How to cite

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Baczyński, Michał, Szostok, Tomasz, and Niemyska, Wanda. "On a functional equation connected to the distributivity of fuzzy implications over triangular norms and conorms." Kybernetika 50.5 (2014): 679-695. <http://eudml.org/doc/262200>.

@article{Baczyński2014,
abstract = {Distributivity of fuzzy implications over different fuzzy logic connectives have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems (see [9, 15] and [4]). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see [5]) \[ f(\min (x+y,a))=\min (f(x)+f(y),b), \] where $a,b>0$ and $f\colon [0,a]\rightarrow [0,b]$ is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation \[ f(m\_1(x+y))=m\_2(f(x)+f(y)), \] where $m_1,m_2$ are functions defined on some intervals of $\{\mathbb \{R\}\}$ satisfying additional assumptions. We analyze the cases when $m_2$ is injective and when $m_2$ is not injective.},
author = {Baczyński, Michał, Szostok, Tomasz, Niemyska, Wanda},
journal = {Kybernetika},
keywords = {fuzzy connectives; fuzzy implication; distributivity; functional equations; fuzzy connectives; fuzzy implication; distributivity; functional equations},
language = {eng},
number = {5},
pages = {679-695},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On a functional equation connected to the distributivity of fuzzy implications over triangular norms and conorms},
url = {http://eudml.org/doc/262200},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Baczyński, Michał
AU - Szostok, Tomasz
AU - Niemyska, Wanda
TI - On a functional equation connected to the distributivity of fuzzy implications over triangular norms and conorms
JO - Kybernetika
PY - 2014
PB - Institute of Information Theory and Automation AS CR
VL - 50
IS - 5
SP - 679
EP - 695
AB - Distributivity of fuzzy implications over different fuzzy logic connectives have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems (see [9, 15] and [4]). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see [5]) \[ f(\min (x+y,a))=\min (f(x)+f(y),b), \] where $a,b>0$ and $f\colon [0,a]\rightarrow [0,b]$ is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation \[ f(m_1(x+y))=m_2(f(x)+f(y)), \] where $m_1,m_2$ are functions defined on some intervals of ${\mathbb {R}}$ satisfying additional assumptions. We analyze the cases when $m_2$ is injective and when $m_2$ is not injective.
LA - eng
KW - fuzzy connectives; fuzzy implication; distributivity; functional equations; fuzzy connectives; fuzzy implication; distributivity; functional equations
UR - http://eudml.org/doc/262200
ER -

References

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