Completely hereditarily atomic OMLs

TL;DR

This paper explores the properties of completely hereditarily atomic orthomodular lattices (OMLs), showing limitations and providing examples of such lattices with specific algebraic and covering properties relevant to quantum logic.

Contribution

It introduces the concept of completely hereditarily atomic OMLs and demonstrates their relation to algebraic and covering properties, extending the framework of quantum predicate logic.

Findings

An irreducible complete atomic OML of infinite height cannot be both algebraic and have the covering property.

Kalmbach's construction yields an algebraic OML with the 2-covering property.

Keller's construction provides a completely hereditarily atomic OML with the covering property.

Abstract

An irreducible complete atomic OML of infinite height cannot both be algebraic and have the covering property. However, Kalmbach's construction provides an example of such an OML that is algebraic and has the 2-covering property, and Keller's construction provides an example of such an OML that has the covering property and is completely hereditarily atomic. Completely hereditarily atomic OMLs generalize algebraic OMLs suitably to quantum predicate logic.