Subalgebras of orthomodular lattices

TL;DR

This paper proves that orthomodular lattices are uniquely determined by their subalgebra structures, extending known results from Boolean algebras and contributing to the foundations of quantum mechanics.

Contribution

It establishes that orthomodular lattices are uniquely identified by their lattice of subalgebras and Boolean subalgebras, generalizing Sachs' Boolean algebra result.

Findings

Orthomodular lattices are determined by their subalgebra lattice.

The poset of Boolean subalgebras also characterizes orthomodular lattices.

Results may impact the foundational approaches to quantum mechanics.

Abstract

Sachs showed that a Boolean algebra is determined by its lattice of subalgebras. We establish the corresponding result for orthomodular lattices. We show that an orthomodular lattice L is determined by its lattice of subalgebras Sub(L), as well as by its poset of Boolean subalgebras BSub(L). The domain BSub(L) has recently found use in an approach to the foundations of quantum mechanics initiated by Butterfield and Isham, at least in the case where L is the orthomodular lattice of projections of a Hilbert space, or von Neumann algebra. The results here may add some additional perspective to this line of work.