Weak tensor products of complete atomistic lattices

Abstract

Abstract.: Given two complete atomistic lattices $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ , we define a set $${\rm S}({\mathcal{L}}_{1}, {\mathcal{L}}_{2})$$ of complete atomistic lattices by means of three axioms (natural regarding the description of separated quantum compound systems), or in terms of a universal property with respect to a given class of bimorphisms. We call the elements of $${\rm S}({\mathcal{L}}_{1}, {\mathcal{L}}_{2})$$ weak tensor products of $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ . We prove that $${\rm S}({\mathcal{L}}_{1}, {\mathcal{L}}_{2})$$ is a complete lattice. We compare the bottom element $${\mathcal{L}}_{1}$$ $${\mathcal{L}}_{2}$$ with the separated product of Aerts and with the box product of Grätzer and Wehrung. Similarly, we compare the top element $${\mathcal{L}}_{1}$$ $${\mathcal{L}}_{2}$$ with the tensor products of Fraser, Chu and Shmuely. With some additional hypotheses on $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ (true for instance if $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$$ are moreover irreducible, orthocomplemented and with the covering property), we characterize the automorphisms of weak tensor products in terms of those of $${\mathcal{L}}_{1}$$ and $${\mathcal{L}}_{2}$