Weak tensor products of complete atomistic lattices
Abstract
Given two complete atomistic lattices L1 and L2, we define a set S(L1,L2) 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(L1,L2) weak tensor products of L1 and L2. We prove that S(L1,L2) is a complete lattice. We compare the bottom element with the separated product of Aerts and with the box product of Graetzer and Wehrung. Similarly, we compare the top element with the tensor products of Fraser, Chu and Shmuely. With some additional hypotheses on L1 and L2 (true for instance if L1 and L2 are moreover irreducible, orthocomplemented and with the covering property), we characterize the automorphisms of weak tensor products in terms of those of L1 and L2.
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