Conditions that Force an Orthomodular Poset to Be a Boolean Algebra

Abstract

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