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

Kód výsledku v 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>

Nalezeny alternativní kódy

RIV/68407700:21230/25:00389455

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

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

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

Výsledek vznikl pri realizaci vícero projektů. Více informací v záložce Projekty.

Návaznosti

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

Rok uplatnění

2025

Kód důvěrnosti údajů

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

Název periodika

Journal of Geometry

ISSN

0047-2468

e-ISSN

1420-8997

Svazek periodika

116

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

CH - Švýcarská konfederace

Počet stran výsledku

24

Strana od-do

33

Kód UT WoS článku

001573939200001

EID výsledku v databázi Scopus

2-s2.0-105016767771