Triangular norm decomposition through methods usingcongruence relations

In this paper, we introduce the concepts of T-distributive element, factor congruence, simple triangular norm and indecomposable t-norm on a bounded lattice through T-compatible congruence and study their related properties. We observe that the smallest t-norm on Boolean algebra is simple. We obtain a congruence by means of T-distributive element and investigate some properties of this type of congruence. A relation between the direct decomposability of any t-norm T and T-distributive elements whose supremum is T-∨-decomposable is obtained. Further, we show that any t-norm on a bounded lattice is directly indecomposable if and only if its underlying bounded lattice is directly indecomposable. We show that any t-norm T on a finite bounded lattice is isomorphic to a direct product of directly indecomposable t-norms. Finally, some illustrative examples are added.