Relatively orthomodular lattices

AbstractM.F. Janowitz defined a generalized orthomodular lattice [GOM-lattice, GOML] as a lattice with a smallest element 0 and with an orthogonality relation ⊥. He proved that any such a lattice is isomorphic to a prime ideal of an orthomodular lattice [OM-lattice, OML]. Equivalently, a GOM-lattice can be defined as a lattice with a smallest element 0, in which every interval containing 0 is an OM-lattice and a natural additional condition is satisfied. We define a relatively orthomodular lattice [ROM-lattice, ROML] as a lattice with a commutativity relation C, in such a way that every GOM-lattice is an ROM-lattice. Equivalently, an ROM-lattice can be defined as a lattice in which every interval is an OM-lattice and a natural additional condition is satisfied. We prove that every ROM-lattice is isomorphic to a prime dual ideal of a GOM-lattice and that this embedding is in some sense minimal. Consequently, an ROM-lattice can be defined as a sublattice of an OM-lattice closed under the relative orthocomplements