Categories of orthogonality spaces

An orthogonality space is a set equipped with symmetric and irreflexive binary relation. We consider spaces the additional property that any collection of mutually orthogonal elements gives rise to structure Boolean algebra. Together maps preserve substructures, we are led category NOS normal spaces. Moreover, an finite rank called linear if for two distinct e f there third one g such exactly pairs e,f e,g have same complement. Linear arise from finite-dimensional Hermitian full subcategory LOS show morphisms orthogonality-preserving lineations. Finally, EOS whose members positive definite over Baer ordered ?-fields Euclidean fixed field. establish induced by generalised semiunitary mappings.