Minimal orthomodular lattices (Q1907548)

For two classes of algebras \(C_2\subseteq C_1\) (minimal) exclusion systems \(\Sigma\subseteq C_1- C_2\) are discussed, for \(C_1\): all orthomodular lattices OML, \(C_2\): all modular ortholattices. A negative answer is given to the question of a finite \(\Sigma\) consisting of finite OML: Every such \(\Sigma\) contains an infinite OML. A minimal OML is a finite nonmodular OML with all proper sub-OML being modular. A characterization of these minimal OML \(T\) is given. Consider the irreducible \(T\) and the equational classes \([T]\) generated by \(T\): Every \([T]\) covers some \([\text{MO}_n]\), \(n\geq 2\), and for fixed \(n\) only finitely many such classes \([T]\) cover \([\text{MO}_n]\). Two such classes \([T]\), \([T']\) are incomparable whenever \(T\), \(T'\) are nonisomorphic.