Coverings of [Mo\(n\)] and minimal orthomodular lattices (Q1264366)

A finite, nonmodular orthomodular lattice (OML) is called minimal if all its proper subOMLs are modular. For a finite, nonmodular OML \(L\), \(L\) is minimal iff the variety generated by \(L\) covers the variety generated by \(\text{MO}n\) for some \(n\geq 2\) (\(\text{MO}n\) denotes the horizontal sum of \(n\) four-element Boolean algebras). Some examples for minimal OMLs are given and pasting-like techniques are described for constructing some new minimal OMLs from certain given ones. By considering the structure of lattices of certain subspaces of three-dimensional vector spaces over certain finite fields one can prove that there exist infinitely many minimal OMLs. Automorphism groups of such OMLs are also described. The paper contains no proofs. Another paper is announced which will contain complete proofs.