Coverings of [MO n ] and minimal orthomodular lattices

Abstract

If T is an orthomodular lattice (OML), we denote by [T ] the equational class generated by T . In this paper we characterize the finite OMLs T such that [T ] covers some [MOn ]. These OMLs T are the non-modular OMLs such that all proper sub-OMLs of T are modular. An OML satisfying that last property is called minimal. There exist infinitely many minimal OMLs provided by quadratic spaces over finite fields. We describe them and give a new way to represent their Greechie diagrams in two separate parts. Other methods to obtain finite minimal OMLs are given.