Constructions of triangular norms on lattices by means of irreducible elements

Constructions and characterizations of triangular norms(t-norms) have been discussed in many different contexts. In this paper, we present two methods to construct t-norms on lattices from given partial information. Since the idea is to follow from part-to-whole, we especially consider the lattices with ∨-decompositions. Firstly, we investigate the structure of the partially ordered set of ∨-irreducible elements. Then, for a given t-norm T on the complete distributive lattice [0, 1]2, we study the restriction of T to the poset of ∨-irreducible elements of [0, 1]2. Furthermore, we give a method for generating t-norms on a finite distributive lattice L by means of ∨-irreducible elements in L. We show that a t-norm on a finite distributive lattice is idempotent if and only if it is idempotent on the set of ∨-irreducible elements. Finally, we introduce a formula to obtain t-norms on L[n] from given t-norms on ∨-irreducible elements of L[n]. We present the dual statements for t-conorms.