Indexed on: 08 Sep '11Published on: 08 Sep '11Published in: Mathematics - Logic
Let L_1 and L_2 be complete atomistic lattices. In a previous paper, we have defined a set S=S(L_1,L_2) of complete atomistic lattices, the elements of which are called weak tensor products of L_1 and L_2. 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 L_1 v L_2, is the tensor product of Fraser whereas the bottom element, denoted by L_1 ^ L_2, is the box product of Graetzer and Wehrung. With some additional hypotheses on L_1 and L_2 (true for instance if L_1 and L_2 are moreover orthomodular with the covering property) we prove that S is a singleton if and only if L_1 or L_2 is distributive, if and only if L_1 v L_2 has the covering property. Our main result reads: L in S admits an orthocomplementation if and only if L=L_1 ^ L_2. At the end, we construct an example in S which has the covering property.