论文标题
有限等级的正交空间的逐渐传递性
Gradual transitivity in orthogonality spaces of finite rank
论文作者
论文摘要
正交性空间与对称和反射性二进制关系在一起。任何配备反身和各向异性内部产品的线性空间都提供一个示例：一维子空间的集合以及通常的正交关系是正交性空间。我们提出了简单的条件，以表征有限维遗传空间以这种方式出现的正交性空间。此外，我们研究了正交性空间允许其任何一对元素之间逐渐过渡的假设的后果。更准确地说，给定元素$ e $和$ f $，我们需要从圆形组的可划分亚组到存在的正交性空间的自动形态群体的同态性，以至于其中一种自动形态图将$ e $ e $ e $ to $ f $，以及任何自动形态的自动形态和$ e $ e $ $ $ e $ f $ f $ f $ e $ e $ f。我们表明我们的假设使我们进入了积极的确定二次空间。通过添加一定的简单条件，我们进一步发现标量的场是Archimedean，因此是真实的子场。
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.
