Hilbert Lattice Equations

TL;DR

This paper explores the relationships among five classes of lattice equations in Hilbert spaces, introduces algorithms for generating specific equations, and clarifies the inclusion relations among these classes.

Contribution

It proposes new algorithms for generating Mayet-Godowski equations and establishes the proper inclusion of the fourth class over the third, advancing understanding of lattice equations in quantum logic.

Findings

The first two classes may coincide.

The fourth class properly includes the third.

An open problem related to the last class is solved.

Abstract

There are five known classes of lattice equations that hold in every infinite dimensional Hilbert space underlying quantum systems: generalised orthoarguesian, Mayet's E_A, Godowski, Mayet-Godowski, and Mayet's E equations. We obtain a result which opens a possibility that the first two classes coincide. We devise new algorithms to generate Mayet-Godowski equations that allow us to prove that the fourth class properly includes the third. An open problem related to the last class is answered. Finally, we show some new results on the Godowski lattices characterising the third class of equations.