A new kind of fuzzy relation equations based on inner transformation

This paper mainly deals with a new kind of fuzzy relation equations based on inner transformation. First, the inner projection of a fuzzy relation and the inner transformation by a fuzzy relation are introduced, and also what is a fuzzy relation equation based on inner transformation is described. Second, the discriminative matrix and the discriminative vector are defined. On the basis of them, an approach of discriminating whether such a fuzzy relation equation has nonzero solution is obtained. At last, a procedure of solving such fuzzy relation equations is presented. The solution set of a fuzzy relation equation based on inner transformation is not closed with respect to join and meet operations, in general. This shows that the traditional methods being used for solving V-Λ composite fuzzy relation equations cannot be applied into the study of this kind of fuzzy relation equations based on inner transformation.