Abstract
An orthogonality space is a set X together with a symmetric and irreflexive binary relation ⊥, called the orthogonality relation. A block partition of X is a partition of a maximal set of mutually orthogonal elements of X, and a decomposition of X is a collection of subsets of X each of which is the orthogonal complement of the union of the others. (X,⊥) is called normal if any block partition gives rise to a unique decomposition of the space. The set of one-dimensional subspaces of a Hilbert space equipped with the usual orthogonality relation provides the motivating example. Together with the maps that are, in a natural sense, compatible with the formation of decompositions from block partitions, the normal orthogonality spaces form a category, denoted by NOS. The objective of the present paper is to characterise both the objects and the morphisms of NOS from various perspectives as well as to compile basic categorical properties of NOS.
Highlights
The motivation underlying the present work is to contribute to a characterisation of Hilbert spaces, the basic models underlying quantum physics
Endowed with the triple relation of linear dependence, P (H) is a projective geometry and taking into account the orthogonality relation, we are led to an orthogeometry [8]
In order to define a category of normal orthogonality spaces, we choose a notion of morphism that takes the formation of decompositions from block partitions in a natural way into account
Summary
The motivation underlying the present work is to contribute to a characterisation of Hilbert spaces, the basic models underlying quantum physics. A further option is to start from structures that were once proposed by D Foulis and his collaborators: a so-called orthogonality space (sometimes referred to as an orthoset) is solely based on a binary relation, assumed to be symmetric and irreflexive. The category of all orthogonality spaces and orthogonality-preserving maps has turned out to be unsuitable In this case an aspect that is central in the quantum-physical formalism is left out of account: the possibility of decomposing a Hilbert space into the direct sum of closed subspaces. In order to define a category of normal orthogonality spaces, we choose a notion of morphism that takes the formation of decompositions from block partitions in a natural way into account. We deal with the decomposition of normal orthogonality spaces into what we call its irreducible subspaces
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