2020/03/06 by Paseka, Jan, Vetterlein, Thomas · 1 citation
#06C15 #46C05 #81P10 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2003.03313
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 ⋆-fields with a Euclidean fixed field. We establish that the morphisms of \mathcal E\mathcal O\mathcal S are induced by generalised semiunitary mappings.