Characterisation of quadratic spaces over the Hilbert field by means of the orthogonality relation.

IF 0.5 Q2 MATHEMATICS
Journal of Geometry Pub Date : 2025-01-01 Epub Date: 2025-09-18 DOI:10.1007/s00022-025-00772-7
Miroslav Korbelář, Jan Paseka, Thomas Vetterlein
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引用次数: 0

Abstract

An orthoset is a set equipped with a symmetric, irreflexive binary relation. With any (anisotropic) Hermitian space H, we may associate the orthoset ( P ( H ) , ) , consisting of the set of one-dimensional subspaces of H and the usual orthogonality relation. ( P ( H ) , ) determines H essentially uniquely.We characterise in this paper certain kinds of Hermitian spaces by imposing transitivity and minimality conditions on their associated orthosets. By gradually considering stricter conditions, we restrict the discussion to a narrower and narrower class of Hermitian spaces. Ultimately, our interest lies in quadratic spaces over countable subfields of R .A line of an orthoset is the orthoclosure of two distinct elements. For an orthoset to be line-symmetric means roughly that its automorphism group acts transitively both on the collection of all lines as well as on each single line. Line-symmetric orthosets turn out to be in correspondence with transitive Hermitian spaces. Furthermore, quadratic orthosets are defined similarly, but are required to possess, for each line , a group of automorphisms acting on transitively and commutatively. We show the correspondence of quadratic orthosets with transitive quadratic spaces over ordered fields. We finally specify those quadratic orthosets that are, in a natural sense, minimal: for a finite n 4 , the orthoset ( P ( R n ) , ) , where R is the Hilbert field, has the property of being embeddable into any other quadratic orthoset of rank n.

Abstract Image

希尔伯特域上二次空间的正交关系表征。
正交集是具有对称的、非自反的二元关系的集合。对于任何(各向异性)厄米空间H,我们可以将正交集(P (H),⊥)与通常的正交关系联系起来,它由H的一维子空间的集合组成。(P (H),⊥)决定了H本质上唯一。本文通过在其相关的正交集上施加传递性和极小性条件,刻画了一类厄米空间。通过逐渐考虑更严格的条件,我们将讨论限制在越来越窄的厄米空间中。最终,我们的兴趣在于R的可数子域上的二次空间。正对集的一条线是两个不同元素的正合。对于直线对称的正交集,大致意味着它的自同构群既作用于所有直线的集合上,也作用于每一条直线上。线对称正交集是与传递厄米空间相对应的。此外,二次正交集的定义与此类似,但要求对每条线具有一组作用于传递和交换的自同构。给出了有序域上具有传递二次空间的二次正交集的对应关系。我们最终指定那些在自然意义上是最小的二次正交集:对于有限的n或4,正交集(P (rn),⊥),其中R是希尔伯特场,具有可嵌入到秩为n的任何其他二次正交集的性质。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of Geometry
Journal of Geometry MATHEMATICS-
CiteScore
1.00
自引率
0.00%
发文量
41
期刊介绍: Journal of Geometry (JG) is devoted to the publication of current research developments in the fields of geometry, and in particular recent results in foundations of geometry, geometric algebra, finite geometries, combinatorial geometry, and special geometries.
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