On the Set-Representable Orthomodular Posets that are Point-Distinguishing

Abstract

Let us denote by \(\mathcal {S}\mathcal {O}\mathcal {M}\mathcal {P}\) the class of all set-representable orthomodular posets and by \(\mathcal {P}\mathcal {D} \mathcal {S}\mathcal {O}\mathcal {M}\mathcal {P}\) those elements of \(\mathcal {S}\mathcal {O}\mathcal {M}\mathcal {P}\) in which any pair of points in the underlying set P can be distinguished by a set (i.e., \((P, \mathcal {L}) \in \mathcal {P}\mathcal {D} \mathcal {S}\mathcal {O}\mathcal {M}\mathcal {P}\) precisely when for any pair \(x, y \in P\) there is a set \(A \in \mathcal {L}\) with \(x \in A\) and \(y \notin A\)). In this note we first construct, for each \((P, \mathcal {L}) \in \mathcal {S}\mathcal {O}\mathcal {M}\mathcal {P}\), a point-distinguishing orthomodular poset that is isomorphic to \((P, \mathcal {L})\). We show that by using a generalized form of the Stone representation technique we also obtain point-distinguishing representations of \((P, \mathcal {L})\). We then prove that this technique gives us point-distinguishing representations on which all two-valued states are determined by points (all two-valued states are Dirac states). Since orthomodular posets may be regarded as abstract counterparts of event structures about quantum experiments, results of this work may have some relevance for the foundation of quantum mechanics.