On Locally Finite Orthomodular Lattices

Abstract

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