Ordinal sums of triangular norms on sublattices of a bounded lattice

Abstract

In this paper, we present several necessary and sufficient conditions to ensure an ordinal sum of t-norms is still a t-norm on bounded lattices. For a fixed or an arbitrary bounded sublattice of a bounded lattice, we provide a direct characterization theorem, respectively. Meanwhile, we extend these results to the ordinal sum for a family of pairwise disjoint bounded sublattices. Furthermore, we investigate ordinal sum of t-norms on product lattices and reveal that there exist ordinal sum t-norms on non-trivial bounded sublattices of product lattices, which is different from the previous results of ordinal sum of t-norms defined on subintervals.