Formal Ordinal Sums of Triangular Norms on Modular Lattices

The formal ordinal sums of t-norms on the bounded lattice is defined by Saminger, but the formal ordinal sum need not be a t-norm. Hence, she provides a necessary and sufficient condition such that the formal ordinal sum is a t-norm indeed. In this paper, we first give several conditions that are equivalent to Saminger’s condition. Secondly, we prove that there is a largest sublattice containing all the pairwise non-overlapped subintervals such that the formal ordinal sum is a t-norm on this sublattice when the bounded lattice is modular.