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

J 2025

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

KORBELÁŘ, Miroslav; Jan PASEKA a Thomas VETTERLEIN

Základní údaje

Originální název

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

Autoři

KORBELÁŘ, Miroslav; Jan PASEKA a Thomas VETTERLEIN

Vydání

Journal of Geometry, Springer Basel AG, 2025, 0047-2468

Další údaje

Jazyk

angličtina

Typ výsledku

Článek v odborném periodiku

Obor

10101 Pure mathematics

Stát vydavatele

Švýcarsko

Utajení

není předmětem státního či obchodního tajemství

Odkazy

URL

Impakt faktor

Impact factor: 0.500 v roce 2024

Označené pro přenos do RIV

Ano

Kód RIV

RIV/00216224:14310/25:00144495

Organizační jednotka

Přírodovědecká fakulta

Klíčová slova anglicky

Hermitian space; Hilbert field; Orthogeometry; Orthogonality space; Orthoset; Projective geometry

Štítky

14311010, rivok

Příznaky

Mezinárodní význam, Recenzováno

Změněno: 13. 3. 2026 14:55, Mgr. Petra Trembecká, Ph.D.

Anotace

V originále

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.

Návaznosti

GF25-20013L, projekt VaV

Název: Ortogonalita a symetrie

Investor: Grantová agentura ČR, Ortogonalita a symetrie, Partnerská agentura

Citovat

KORBELÁŘ, Miroslav; Jan PASEKA a Thomas VETTERLEIN. Characterisation of quadratic spaces over the Hilbert field by means of the orthogonality relation. Journal of Geometry. Springer Basel AG, 2025, roč. 116, č. 3, s. 33-56. ISSN 0047-2468. Dostupné z: https://doi.org/10.1007/s00022-025-00772-7.

@article{2529177,
   author = {Korbelář, Miroslav and Paseka, Jan and Vetterlein, Thomas},
   article_number = {3},
   doi = {https://doi.org/10.1007/s00022-025-00772-7},
   keywords = {Hermitian space; Hilbert field; Orthogeometry; Orthogonality space; Orthoset; Projective geometry},
   language = {eng},
   issn = {0047-2468},
   journal = {Journal of Geometry},
   title = {Characterisation of quadratic spaces over the Hilbert field by means of the orthogonality relation},
   url = {https://link.springer.com/article/10.1007/s00022-025-00772-7},
   volume = {116},
   year = {2025}
}

TY  - JOUR
ID  - 2529177
AU  - Korbelář, Miroslav - Paseka, Jan - Vetterlein, Thomas
PY  - 2025
TI  - Characterisation of quadratic spaces over the Hilbert field by means of the orthogonality relation
JF  - Journal of Geometry
VL  - 116
IS  - 3
SP  - 33
EP  - 33
PB  - Springer Basel AG
SN  - 00472468
KW  - Hermitian space
KW  - Hilbert field
KW  - Orthogeometry
KW  - Orthogonality space
KW  - Orthoset
KW  - Projective geometry
UR  - https://link.springer.com/article/10.1007/s00022-025-00772-7
N2  - 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.
ER  -

KORBELÁŘ, Miroslav; Jan PASEKA a Thomas VETTERLEIN. Characterisation of quadratic spaces over the Hilbert field by means of the orthogonality relation. \textit{Journal of Geometry}. Springer Basel AG, 2025, roč.~116, č.~3, s.~33-56. ISSN~0047-2468. Dostupné z: https://doi.org/10.1007/s00022-025-00772-7.

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