Completely hereditarily atomic OMLs

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.