Antitone involutions on tensor products of complete lattices and complete distributivity

Abstract

The purpose of this paper is to study antitone involutions on tensor products of complete lattices. A lattice with antitone involution is called an involution lattice. We show that if M is a completely distributive involution lattice, then for each complete involution lattice L there exists a unique antitone involution on the tensor product $M\otimes L$ such that the natural embeddings of M and L into $M\otimes L$ are involution-preserving. This is best possible, since the described property characterizes complete distributivity in the class of complete involution lattices. When M and L are completely distributive involution lattices, $M\otimes L$ with the aforementioned antitone involution is the codomain of a universal bimorphism in the sense of the category of all completely distributive de Morgan algebras and their join- and involution-preserving maps. The case that M and L are orthocomplemented is explored too.