On two functional equations connected with distributivity of fuzzy implications

• Autorzy: Roman Ger , Marcin Emil Kuczma , Wanda Niemyska
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

fuzzy implication; distributivity; functional equation; t-conorm

• Wydawca: Polish Mathematical Society
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2015
• Tom: 55
• Numer: 2

Daty

wydano

2015

online

2016-05-25

Twórcy

autor

Roman Ger

autor

Marcin Emil Kuczma

autor

Wanda Niemyska

Identyfikatory

DOI

10.14708/cm.v55i2.1118