Infinitely ∨-distributive t-norms on complete lattices and pseudo-complements

In this paper, we have introduced the concepts of T-prime element, T-prime radical element and T-semi-prime element and have studied their properties. We have investigated the construction methods of new infinitely ∨-distributive t-norms from given infinitely ∨-distributive t-norms by means of T-prime elements and pseudo-complements. If every element of L is T-prime radical element, then we show that L is a complete Brouwerian lattice and T=∧. If L is an algebraic lattice, T is a t-norm on L and every compact element of L is idempotent, then we obtain that T=∧. We show that LR is a complete lattice and that the restriction of T to LR is an infinitely ∪-distributive t-norm on LR. When T is an infinitely ∨-distributive t-norm on L and 0 is a T-prime, we present infinitely ∨-distributive t-norms on a complete sublattice L* and study their properties.