Publication details

Categories of orthogonality spaces



Year of publication 2022
Type Article in Periodical
Magazine / Source Journal of Pure and Applied Algebra
MU Faculty or unit

Faculty of Science

Keywords Orthogonality spaces; Undirected graphs; Categories; Boolean subalgebras; Linear orthogonality spaces; Generalised semilinear map
Description 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 NOS 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 LOS of NOS and we show that the morphisms are the orthogonality-preserving lineations. Finally, we consider the full subcategory OS pound of LOS whose members arise from positive definite Hermitian spaces over Baer ordered *-fields with a Euclidean fixed field. We establish that the morphisms of OS pound are induced by generalised semiunitary mappings.
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