∨-Distributive and infinitely ∨-distributive t-norms on complete lattices

A relation between the direct decomposability of an infinitely ∨-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 T1-lattices without infinitely ∨-distributive t-norms on complete lattices and the description of all ∨-distributive t-norms on lattices of length 2 are given. The paper also contains the solutions of two open problems.