Distributive Equations of Implications Based on Continuous Triangular Norms (I)

作者：Qin, Feng^*; Baczynski, Michal; Xie, Aifang

来源：IEEE Transactions on Fuzzy Systems, 2012, 20(1): 153-167.

DOI：10.1109/TFUZZ.2011.2171188

• Archimedean t-norm
• combs methods
• continuous t-norm
• functional equations
• fuzzy implication

In order to avoid combinatorial rule explosion in fuzzy reasoning, in this paper, we explore the distributive equations of implications. In detail, by means of the sections of I, we give out the sufficient and necessary conditions of solutions for the distributive equation of implication I(x, T-1 (y, z)) = T-2 (I(x, y), I(x, z)), when T-1 is a continuous but not Archimedean triangular norm, T-2 is a continuous and Archimedean triangular norm, and I is an unknown function. This obtained characterizations indicate that there are no continuous solutions for the previous functional equation, satisfying the boundary conditions of implications. However, under the assumptions that I is continuous except for the point (0,0), we get its complete characterizations. Here, it should be pointed out that these results make differences with recent results that are obtained by Baczynski and Qin. Moreover, our method can still apply to the three other functional equations that are related closely to the distributive equation of implication.

• 出版日期2012-2
• 单位南昌航空大学; 江西师范大学; 山东大学