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 structures, 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 $\star$-fields with a Euclidean fixed field. We establish that the morphisms of ${\mathcal E}{\mathcal O}{\mathcal S}$ are induced by generalised semiunitary mappings.