On the distributivity of fuzzy implications over continuous and Archimedean triangular conorms
This paper essentially proposes a mild generalization of the distributive functional equation \[ I((x,S_1(y,z))=S_2(I(x,y),I(x,z)),\quad x,y,z\in\,[0,1], \] where \(I\) is any function and \(S_1,S_2\) are t-conorms, already solved when \(S_1\) and \(S_2\) are both strict or both nilpotent, to the case when one is strict and the other one nilpotent. Section 2 contains only basic notations and definitions. The investigation is conducted in the same style as in the cited work numbered [6], exploiting some results, contained in Section 3, pertaining to the famous, additive Cauchy functional equation. Some remarks should be made with regard to the proof of Proposition 3.4: third line of this proof, page 1409, ``both sides of (5)'' instead of ``both sides of (6)'' and ``there exists \(x_1\in\,\,]x_0,x]\)'' instead of ``there exists \(x_1\in\,[x_0,x]\)'' on page 1410, just after formula (11), but mostly a passage is wrong, in the case \(x_0=\infty\), which begins seven lines from the bottom of page 1410. In fact, to cover this case, the author invokes Theorem 3.3, with \(a=x_0\), but in this theorem \(a\) is explicitly assumed to be a real, finite, positive number and not \(\infty\). I suppose that the proof should work in the following way: we show that, if \(x_0=\infty\), then \(f(x)=0\) for all \(x\geq 0\). If not, let \(x_1\) be a positive number such that \(0<f(x_1)<b\). Then, \(f(2x_1)=\min\{f(x_1)+f(x_1),b\}<b\) (note that \(f(2x_1)=b\) is incompatible with the nature of \(x_0\)), hence \(2f(x_1)<b\) and, by induction, we easily get \(nf(x_1)<b\) for any \(n\in \mathbb N\) which clearly contradicts \(f(x_1)>0\).
scientific article; zbMATH DE number 5705389