New construction of t-norms and t-conorms on bounded lattices

Abstract

This article studies the construction of ordinal sums of triangular norms (t-norms) and triangular conorms (t-conorms) on bounded lattices. First, we present a new class of t-norms and t-conorms on an arbitrary bounded lattice. Some examples are provided to illustrate that our new construction method is different from some existing methods for the construction of ordinal sums of t-norms and t-conorms on an arbitrary bounded lattice. Then we illustrate that our new construction method can be generalized by induction to a modified ordinal sum construction for t-norms and t-conorms on an arbitrary bounded lattice. Finally, for any bounded sublattice S of a bounded lattice L, we provide a sufficient condition that does not need S and L to have the same bottom and top elements such that an ordinal sum function is a t-norm on L, whereas the function is determined by an arbitrary selection of bounded sublattices as carriers for arbitrary summand t-norms. In particular, we give a necessary and sufficient condition that needs S to have the same bottom element as L such that an ordinal sum function is a t-norm on L.