Weak tensor products of complete atomistic lattices

Given two complete atomistic lattices L-1 and L-2, we define a set S (L-1, 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 S(L-1, L-2) weak tensor products of L-1 and L-2. We prove that S(L-1, L-2) is a complete lattice. We compare the bottom element L1L2 with the separated product of Aerts and with the box product of Gratzer and Wehrung. Similarly, we compare the top element L1L2 with the tensor products of Fraser, Chu and Shmuely. With some additional hypotheses on L, and L-2 (true for instance if Ll and L-2 are moreover irreducible, orthocomplemented and with the covering property), we characterize the automorphisms of weak tensor products in terms of those of L-1 and L-2.