A relation between the direct decomposability of an infinitely v-distributive t-norm on a complete lattice L and direct decompositions of the neutral element 1 of L is obtained. Some useful applications of this relation are given. In particularly, the description of all T-1-lattices without infinitely V-distributive t-norms on complete lattices and the description of all V-distributive t-norms on lattices of length 2 are given. The paper also contains the solutions of two open problems. (C) 2004 Elsevier B.V. All rights reserved.