Orthocomplemented weak tensor products
Abstract
Let L1 and L2 be complete atomistic lattices. In a previous paper, we have defined a set S=S(L1,L2) of complete atomistic lattices, the elements of which are called weak tensor products of L1 and L2. S is defined by means of three axioms, natural regarding the description of some compound systems in quantum logic. It has been proved that S is a complete lattice. The top element of S, denoted by L1 v L2, is the tensor product of Fraser whereas the bottom element, denoted by L1 L2, is the box product of Graetzer and Wehrung. With some additional hypotheses on L1 and L2 (true for instance if L1 and L2 are moreover orthomodular with the covering property) we prove that S is a singleton if and only if L1 or L2 is distributive, if and only if L1 v L2 has the covering property. Our main result reads: L in S admits an orthocomplementation if and only if L=L1 L2. At the end, we construct an example in S which has the covering property.
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