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

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>I</i>, we give out the sufficient and necessary conditions of solutions for the distributive equation of implication <i>I</i>(<i>x</i>,<i>T</i>₁(<i>y</i>,<i>z</i>))=<i>T</i>₂(<i>I</i>(<i>x</i>,<i>y</i>),<i>I</i>(<i>x</i>,<i>z</i>)), when <i>T</i>₁ is a continuous but not Archimedean triangular norm, <i>T</i>₂ is a continuous and Archimedean triangular norm, and <i>I</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>I</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 Baczyński and Qin. Moreover, our method can still apply to the three other functional equations that are related closely to the distributive equation of implication.