Coverings of [MO n ] and minimal orthomodular lattices

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 MO _n ]. 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. Received May 14, 2005; accepted in final form May 30, 2007.