Lattice-based Sum Construction of Nullnorms on Bounded Lattices

Abstract

Nullnorms on bounded lattices are generalizations of t-norms and t-conorms with a zero element laying anywhere in the underlying lattices. In this paper, new methods for constructing nullnorms on bounded lattices are proposed. The proposed construction methods are based on the lattice-based sum of lattices that has been recently introduced by El-Zekey et al. (see [9]), for building new (bounded) lattices from fixed ones indexed by a (finite) lattice-ordered index set. Subsequently, the new construction methods are applied for building several new families of nullnorms on bounded lattices. As a by-product, lattice-based sum constructions of t-norms and t-conorms on bounded lattices have been obtained. Furthermore, new idempotent nullnorms on bounded lattices have been also obtained.