Reconstructing an atomic orthomodular lattice from the poset of its Boolean sublattices

TL;DR

This paper demonstrates that atomic orthomodular lattices can be uniquely reconstructed from their Boolean sublattice posets, providing insights relevant to quantum theory and the topos approach.

Contribution

It introduces a method to reconstruct atomic orthomodular lattices from the poset of their Boolean subalgebras, advancing the understanding of lattice structures in quantum theory.

Findings

Reconstruction of atomic orthomodular lattices from Boolean sublattice posets

Application to quantum theory and topos approach

Insight into the structure of projection lattices in Hilbert spaces

Abstract

We show that an atomic orthomodular lattice L can be reconstructed up to isomorphism from the poset B(L) of Boolean subalgebras of L. A motivation comes from quantum theory and the so-called topos approach, where one considers the poset of Boolean sublattices of L=P(H), the projection lattice of the algebra B(H) of bounded operators on Hilbert space.