Some characterizations of the tensor product of complete lattices with applications to quantales

Abstract

This paper examines tensor products of complete lattices in which one factor is com- pletely distributive. At least five characterizations of complete distributivity involving tensor products of complete lattices are given, among them this one: M is a completely distributive lattice if and only if for every complete lattice L the tensor product M ⊗ L is order isomorphic to the partially ordered set of all join- and meet-reversing maps from the complete lattice of all upclosed subsets of L to the lattice M. Some of these characterizations are then applied to give explicit descriptions of the multiplication of the tensor product of two quantales one of which is completely distributive.