On two functional equations connected with distributivity of fuzzy implications

Roman Ger ; Marcin Emil Kuczma ; Wanda Niemyska

Commentationes Mathematicae, Tome 55 (2015) no. 2, p. 163−169.

Résumé

The distributivity law for a fuzzy implication \(I\colon [0,1]^2 \to [0,1]\) with respect to a fuzzy disjunction \(S\colon [0,1]^2 \to [0,1]\) states that the functional equation \( I(x,S(y,z))=S(I(x,y),I(x,z)) \) 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: \( h(\min(xg(y),1)) = \min(h(x)+ h(xy),1)\), \(x \in (0,1)\), \(y \in (0,1]\), and \( h(xg(y)) = h(x)+ h(xy)\), \(x,y \in (0, \infty)\), in the class of increasing bijections \(h\colon [0,1] \to [0,1]\) with an increasing function \(g\colon (0,1] \to [1, \infty)\) and in the class of monotonic bijections \(h\colon (0, \infty) \to (0, \infty)\) with a function \(g\colon (0, \infty) \to (0, \infty)\), respectively. A description of solutions in more general classes of functions (including nonmeasurable ones) is presented.

Détail

DOI : 10.14708/cm.v55i2.1118

Classification : 39B22, 94D05
Mots-clés : fuzzy implication, distributivity, functional equation, t-conorm