Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gradual transitivity in orthogonality spaces of finite rank   An orthogonality space is a set together with a symmetric and irreflexive
binary relation. Any linear space equipped with a reflexive and anisotropic
inner product provides an example: the set of one-dimensional subspaces
together with the usual orthogonality relation is an orthogonality space. We
present simple conditions to characterise the orthogonality spaces that arise
in this way from finite-dimensional Hermitian spaces. Moreover, we investigate
the consequences of the hypothesis that an orthogonality space allows gradual
transitions between any pair of its elements. More precisely, given elements
$e$ and $f$, we require a homomorphism from a divisible subgroup of the circle
group to the automorphism group of the orthogonality space to exist such that
one of the automorphisms maps $e$ to $f$, and any of the automorphisms leaves
the elements orthogonal to $e$ and $f$ fixed. We show that our hypothesis leads
us to positive definite quadratic spaces. By adding a certain simplicity
condition, we furthermore find that the field of scalars is Archimedean and
hence a subfield of the reals.

5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph
37
The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
15
The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin