Convolutions of Lattice Functions Based on Triangular Norms and Conorms

This article aims to investigate the convolutions on the set of lattice functions between two bounded lattices based on triangular norms (t-norms), which is also a generalization of t-norms on bounded lattices to convolution lattices. It first shows that the set of all convex and idempotent functions with an extended t-norm is a commutative po-monoid. And then, one gains that an extended t-norm is a lattice-ordered t-norm on convolution lattices when the t-norm on domain lattices satisfies the intermediate value property. Finally, one derives analogous results for the convolutional triangular conorms (t-conorms) by the duality between t-norms and t-conorms. One anticipates that these results can be utilized in many-valued logic and in several branches of information sciences and mathematics that deal with algebraic structures.