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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]) $$f(min(x+y,a\left)\right)=min\left(f\right(x)+f(y),b),$$ where $a,b>0$ and $f:[0,a]\to [0,b]$ is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation $$f\left(m_1(x+y)\right)=m_2(f\left(x\right)+f\left(y\right)),$$ where $m_1,m_2$ are functions...

The distributivity law for a fuzzy implication $I:{[0,1]}^2\to [0,1]$ with respect to a fuzzy disjunction $S:{[0,1]}^2\to [0,1]$ states that the functional equation $I(x,S(y,z\left)\right)=S\left(I\right(x,y),I(x,z\left)\right)$ is satisfied for all pairs $(x,y)$ from the unit square. To compare some results obtained while solving this equation in various classes of fuzzy implications, Wanda Niemyska has reduced the problem to the study of the following two functional equations: $h(min(xg\left(y\right),1\left)\right)=min\left(h\right(x)+h(xy),1)$, $x\in (0,1)$, $y\in (0,1]$, and $h\left(xg\right(y\left)\right)=h\left(x\right)+h\left(xy\right)$, $x,y\in (0,\infty )$, in the class of increasing bijections $h:[0,1]\to [0,1]$ with an increasing function $g:(0,1]\to [1,\infty )$ and in the class of monotonic bijections...