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

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F25%3A00144495" target="_blank" >RIV/00216224:14310/25:00144495 - isvavai.cz</a>

Alternative codes found

RIV/68407700:21230/25:00389455

Result on the web

<a href="https://link.springer.com/article/10.1007/s00022-025-00772-7" target="_blank" >https://link.springer.com/article/10.1007/s00022-025-00772-7</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00022-025-00772-7" target="_blank" >10.1007/s00022-025-00772-7</a>

Result language

angličtina

Original language name

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

Original language description

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(Rn),⊥), where R is the Hilbert field, has the property of being embeddable into any other quadratic orthoset of rank n.

Czech name

—

Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10101 - Pure mathematics

Project

Result was created during the realization of more than one project. More information in the Projects tab.

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2025

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Journal of Geometry

ISSN

0047-2468

e-ISSN

1420-8997

Volume of the periodical

116

Issue of the periodical within the volume

3

Country of publishing house

CH - SWITZERLAND

Number of pages

24

Pages from-to

33

UT code for WoS article

001573939200001

EID of the result in the Scopus database

2-s2.0-105016767771