On two functional equations connected with distributivity of fuzzy implications

• Autorzy: Ger R. , Kuczma M. E. , Niemyska W.

Identyfikatory

DOI

10.14708/cm.v55i2.1118

• Języki publikacji: EN

Abstrakty

EN

The distributivity law for a fuzzy implication I:[0,1]²→[0,1] with respect to a fuzzy disjunction S:[0,1]²→[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∈(0,1), y∈(0,1], and h(xg(y))=h(x)+h(xy), x,y∈(0,∞), in the class of increasing bijections h:[0,1]→[0,1] with an increasing function g:(0,1]→[1,∞) and in the class of monotonic bijections h:(0,∞)→(0,∞) with a function g:(0,∞)→(0,∞), 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: Polskie Towarzystwo Matematyczne
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2015
• Tom: Vol 55, No. 2
• Strony: 163--169
• Opis fizyczny: Bibliogr. 4 poz.

Twórcy

autor

Ger R.

• Institute of Mathematics Silesian University, Bankowa 14, 40-007 Katowice, Poland

autor

Kuczma M. E.

• Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-013 Warsaw, Poland

autor

Niemyska W.

• Institute of Mathematics Silesian University, Bankowa 14, 40-007 Katowice, Poland

Bibliografia

• [1] M. Baczyński and B. Jayaram, On the distributivity of fuzzy implications over nilpotent or strict triangular conorms, IEEE Trans. Fuzzy Syst. 17 (June 2009), no. 3,590-603, DOI 10.1109/tfazz.2008.924201.
• [2] J. Balasubramaniam and C. J. M. Rao, On the distributivity of implication operators over T and S norms, IEEE Trans. Fuzzy Syst. 12 (April 2004), no. 2,194-198, DOI 10.1109/tfuzz.2004.825075.
• [3] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Państwowe Wydawnictwo Naukowe 8c Uniwersytet Śląski, Warszawa-Kraków-Katowice 1985, DOI 10.1007/978-3-7643-8749-5; second edition: Birkhäuser Verlag, Basel-Boston-Berlin, 2009.
• [4] W. Niemyska, On functional equations connected with the distributivity of fuzzy implications, Katowice 2015; in Polish.