Conditions that Force an Orthomodular Poset to Be a Boolean Algebra

Abstract

We introduce two new classes of orthomodular posets—the class of weakly Boolean orthomodular posets and the class of orthomodular posets with the property of maximality. The main result of this paper is that the intersection of these classes is the class of Boolean algebras. Since the first class introduced here contains various classes of orthomodular posets with a given property of its state space and the second class contains, e.g., lattice (orthocomplete, resp.) orthomodular posets, the main theorem can be viewed as a generalization of various results concerning the question when an orthomodular poset has to be a Boolean algebra. Moreover, it gives alternative proofs to previous results and new results of this type.