ON LOCALLY FINITE ORTHOMODULAR LATTICES

Let us denote by GF the class of all orthomodular lattices (OMLs) that are locally finite (i.e., L ? GF 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 GF. We find GF considerably large though, obviously, not all OMLs belong to GF. Then we study states on the OMLs of GF. We show that local finiteness may to a certain extent make up for distributivity. For instance, we show that if L ? GF 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 GF relevant to the quantum logic theory.