Distributivity equations of implications based on continuous triangular conorms (Ⅱ)

摘要

In order to avoid combinatorial rule explosion in fuzzy reasoning, Qin and Baczynski, in , investigated the distributivity 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 / is an unknown function. In fact, it partially answered the open problem suggested by Baczynski and Jayaram in . In this work we continue to explore the distributivity equation of implication I(x, S_1 (y, z)) = S_2(I(x, y), I(x, z)), when both S_1 and S_2 are continuous but not Archimedean triangular conorms, and I is an unknown function. Here it should be pointed out that these results make difference with recent ones obtained in . Moreover, our method can still apply to the three other functional equations related closely to this equation. It is in this sense that we have completely solved the open problem commented above.