Remarks on Concrete Orthomodular Lattices

An orthomodular lattice (OML) is called concrete if it is isomorphic to a collection of subsets of a set with partial ordering given by set inclusion, orthocomplementation given by set complementation, and finite orthogonal joins given by disjoint unions. Interesting examples of concrete OMLs are obtained by applying Kalmbach’s construction K (L) to an arbitrary bounded lattice L . This note provides several results regarding Kalmbach’s construction, concrete OMLs, and the relationship between the notions. First, we provide order-theoretic and categorical characterizations of the OML K (L) in terms of the bounded lattice L . Second, we provide an identity satisfied by each OML K (L), but not valid in every concrete OML. This shows that the class of OMLs of the form K (L) do not generate the variety of all concrete OMLs. Finally, we show that every concrete OML can be embedded into a concrete OML in which every element is a join of two or fewer atoms.