Orthocomplemented Posets with a Symmetric Difference

Abstract

We endow orthocomplemented posets with a binary operation–an abstract symmetric difference of sets–and we study algebraic properties of this class, ${\cal ODP}$ . Denoting its elements by ODP, we first investigate on the features related to compatibility in ODPs. We find, among others, that any ODP is orthomodular. This explicitly links ${\cal ODP}$ with the theory of quantum logics. By analogy with Boolean algebras, we then ask if (when) an ODP is set representable. Though we find that general ODPs do not have to be set representable, many natural ODPs are shown to be. We characterize the set-representable ODPs in terms of two valued morphisms and prove that they form a quasivariety. This quasivariety contains the class of pseudocomplemented ODPs as we show afterwards. At the end we ask whether any orthomodular poset can be converted or, more generally, enlarged to an ODP. By countre-examples we answer these questions to the negative.