Ordinal Sum of Two Binary Operations Being a T-Norm on Bounded Lattice

The ordinal sum of t-norms on a bounded lattice has been used to construct other t-norms. However, an ordinal sum of binary operations (not necessarily t-norms) defined on the fixed subintervals of a bounded lattice may not be a t-norm. Some necessary and sufficient conditions are presented in this article for ensuring that an ordinal sum on a bounded lattice of two binary operations is, in fact, a t-norm. In particular, the results presented here provide an answer to an open problem put forward by Ertuğrul and Yeşilyurt,
ordinal sums of triangular norms on bounded lattices
, Inf. Sci., vol. 517, (2020) 198–216.