Orthocomplemented weak tensor products

TL;DR

This paper investigates the structure of weak tensor products of complete atomistic lattices, establishing conditions for orthocomplementation and exploring their properties in quantum logic contexts.

Contribution

It characterizes when elements in the set of weak tensor products admit orthocomplementation, linking it to the bottom element L_1 ^ L_2, and provides examples with the covering property.

Findings

L admits an orthocomplementation iff L=L_1 ^ L_2.

S is a singleton iff L_1 or L_2 is distributive.

An example in S with the covering property is constructed.

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.