Rules of Partial Orthomodularity

The rule of orthomodularity is important, for example due to its foundational role in quantum logic. However, orthomodularity is a restriction that is not always suitable. For example, orthomodularity is not fulfilled by certain geometric structures such as the lattice of closed convex cones, which is different from the lattice considered in quantum logic, namely that of closed subspaces of a Hilbert space. Thus, the question arises whether the rule of orthomodularity can be relaxed such that the relaxed rule is still stronger than the ortholattice rule and such that each orthomodular lattice fulfills also the relaxed rule. Therefore, we present two rules of partial orthomodularity of different strength, (pOM) and (pOM\(_{ex}\)), which keep some of the advantages of the rule of orthomodularity but are relaxations of it. We show the validity and usefulness of these rules by proving a subalgebra theorem for (pOM), by showing that an algebraic representation theorem for orthomodularity can be relaxed as to be based on the rules of partial orthomodularity and by proving a connection to Johansson’s minimal rule.