On locally finite orthomodular lattices

TL;DR

This paper explores the class of locally finite orthomodular lattices, showing how to generate new ones, analyzing their properties, and examining states relevant to quantum logic, highlighting how local finiteness influences distributivity and state existence.

Contribution

It introduces methods to construct new locally finite orthomodular lattices and studies their algebraic and state properties, advancing understanding in quantum logic.

Findings

Locally finite OMLs can be expanded from initial ones.

Local finiteness can compensate for lack of distributivity.

Existence of states on finite subOMLs implies a state on the entire lattice.

Abstract

Let us denote by LF the class of all orthomodular lattices (OMLs) that are locally finite (i.e., L in LF provided each finite subset of L generates in L a finite subOML). We first show in this note how one can obtain new locally finite OMLs from the initial ones and enlarge thus the class LF . We find LF considerably large though, obviously, not all OMLs belong to LF . We then study states on the OMLs of LF . We show that local finiteness may to a certain extent make up for distributivity. We for instance show that if L in LF and if for any finite subOML K there is a state s : K to [0, 1] on K, then there is a state on the entire L. We also consider further algebraic and state properties of LF relevant to quantum logic theory.