Geodesic
On a functional equation connected to the distributivity of fuzzy implications over triangular norms and conorms
Kybernetika, Tome 50 (2014) no. 5, pp. 679-695.

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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]) $$ f(\min(x+y,a))=\min(f(x)+f(y),b), $$ 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 ${\mathbb R}$ satisfying additional assumptions. We analyze the cases when $m_2$ is injective and when $m_2$ is not injective.
DOI : 10.14736/kyb-2014-5-0679
Classification : 03B52, 03E72, 39B05, 39B22, 39B99
Keywords: fuzzy connectives; fuzzy implication; distributivity; functional equations 
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     title = {On a functional equation connected to the distributivity of fuzzy implications over triangular norms and conorms},
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Baczyński, Michał; Szostok, Tomasz; Niemyska, Wanda. On a functional equation connected to the distributivity of fuzzy implications over triangular norms and conorms. Kybernetika, Tome 50 (2014) no. 5, pp. 679-695. doi : 10.14736/kyb-2014-5-0679. http://geodesic.mathdoc.fr/articles/10.14736/kyb-2014-5-0679/

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