Distributive equations of implications based on nilpotent triangular norms

Authors:
Feng Qin;Li Yang
Affiliations:
College of Mathematics and Information Science, Nanchang Hangkong University, 330063 Nanchang, PR China and College of Mathematics and Information Science, Jiangxi Normal University, 330022 Nancha ...;College of Mathematics and Information Science, Jiangxi Normal University, 330022 Nanchang, PR China
Venue:
International Journal of Approximate Reasoning
Year:
2010

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Abstract

In this paper, we explore the distributive equations of implications, both independently and along with other equations. In detail, we consider three classes of equations. (1) By means of the section of I, we give out the sufficient and necessary conditions of solutions for the distributive equation of implication I(x,T(y,z))=T(I(x,y),(x,z)) based on a nilpotent triangular norm T and an unknown function I, which indicates that there are no continuous solutions satisfying the boundary conditions of implications. Under the assumptions that I is continuous except the vertical section I(0,y), y@?[0,1), we get its complete characterizations. (2) We prove that there are no solutions for the functional equations I(x,T(y,z))=T(I(x,y),I(x,z)),I(x,I(y,z))=I(T(x,y),z). (3) We obtain the sufficient and necessary conditions on T and I to be solutions of the functional equations I(x,T(y,z))=T(I(x,y),I(x,z)), I(x,y)=I(N(y),N(x)).