Categories of orthogonality spaces

Abstract

An orthogonality space is a set equipped with a symmetric and irreflexive binary relation. We consider orthogonality spaces with the additional property that any collection of mutually orthogonal elements gives rise to the structure of a Boolean algebra. Together with the maps that preserve the Boolean substructures, we are led to the category $\mathcal{N}\mathcal{O}\mathcal{S}$ of normal orthogonality spaces.

Moreover, an orthogonality space of finite rank is called linear if for any two distinct elements e and f there is a third one g such that exactly one of f and g is orthogonal to e and the pairs $e,f$ and $e,g$ have the same orthogonal complement. Linear orthogonality spaces arise from finite-dimensional Hermitian spaces. We are led to the full subcategory $\mathcal{L}\mathcal{O}\mathcal{S}$ of $\mathcal{N}\mathcal{O}\mathcal{S}$ and we show that the morphisms are the orthogonality-preserving lineations.

Finally, we consider the full subcategory $\mathcal{E}\mathcal{O}\mathcal{S}$ of $\mathcal{L}\mathcal{O}\mathcal{S}$ whose members arise from positive definite Hermitian spaces over Baer ordered ⋆-fields with a Euclidean fixed field. We establish that the morphisms of $\mathcal{E}\mathcal{O}\mathcal{S}$ are induced by generalised semiunitary mappings.