In this paper, we define the set of incomparable elements with respect to the triangular order for any t-norm on a bounded lattice. By means of the triangular order, an equivalence relation on the class of t-norms on a bounded lattice is defined and this equivalence is deeply investigated. Finally, we discuss some properties of this equivalence.
For a t-norm T on a bounded lattice (L,≤), a partial order ≤T was recently defined and studied. In \cite{Karacal11}, it was pointed out that the binary relation ≤T is a partial order on L, but (L,≤T) may not be a lattice in general. In this paper, several sufficient conditions under which (L,≤T) is a lattice are given, as an answer to an open problem posed by the authors of \cite{Karacal11}. Furthermore, some examples of t-norms on L such that (L,≤T) is a lattice are presented.