Kybernetika 50 no. 5, 679-695, 2014

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

Michał Baczyński, Tomasz Szostok and Wanda NiemyskaDOI: 10.14736/kyb-2014-5-0679

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 \cite{Combs_Andrews_1998,Jayaram2008} and \cite{Baczynski_Jayaram_2008}). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see \cite{Baczynski_Jayaram_2009}) $$ f(\min(x+y,a))=\min(f(x)+f(y),b), $$ where $a,b>0$ and $f\colon[0,a]\to[0,b]$ 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 $\R$ satisfying additional assumptions. We analyze the cases when $m_2$ is injective and when $m_2$ is not injective.

Keywords:

fuzzy implication, distributivity, fuzzy connectives, functional equations

Classification:

03B52, 03E72, 39B99

paper.pdf

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