Lattice-based sum of t-norms on bounded lattices

Abstract

The concept of ordinal sums in the sense of Clifford have long been blamed for their limitations in constructing new t-norms including inability to cope with general bounded lattices. Motivated by this observation, and based on the lattice-based sum of lattices that has been recently introduced by El-Zekey et al., we propose a new sum-type construction of t-norms, called a lattice-based sum of t-norms, for building new t-norms on bounded lattices from given ones. The proposed sum is generalizing the well-known ordinal and horizontal sum constructions of t-norms by allowing for lattice ordered index sets. We demonstrate that, like the ordinal sum of t-norms, the lattice-based sum of t-norms can be generalized using as summands so-called t-subnorms, still leading to a t-norm. Subsequently, we apply the results for constructing several new families of t-norms and t-subnorms on bounded lattices. In the same spirit, by the duality, we will also introduce lattice-based sums of t-conorms and t-subconorms.