In this study, we introduce new methods for constructing t-norms and t-conorms on a bound\-ed lattice L based on a priori given t-norm acting on [a,1] and t-conorm acting on [0,a] for an arbitrary element a∈L∖{0,1}. We provide an illustrative example to show that our construction methods differ from the known approaches and investigate the relationship between them. Furthermore, these methods are generalized by iteration to an ordinal sum construction for t-norms and t-conorms on a bounded lattice.
In this paper, an equivalence on the class of uninorms on a bounded lattice is discussed. Some relationships between the equivalence classes of uninorms and the equivalence classes of their underlying t-norms and t-conorms are presented. Also, a characterization for the sets admitting some incomparability w.r.t. the U-partial order is given.
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.