# Orthocomplemented weak tensor products

Research paper by **Boris Ischi**

Indexed on: **08 Sep '11**Published on: **08 Sep '11**Published in: **Mathematics - Logic**

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#### Abstract

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.