Triangular norm decompositions through methods using congruence relations. Issue 3 (3rd April 2022) - Karaçal, Funda; Arpacı, Samet; Karacair, Kübra
Abstract : 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.