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

TL;DR

The paper develops a combinatorial, symmetry-driven path to characterise quadratic spaces over the Hilbert field via their orthogonality relations on one-dimensional subspaces. By introducing line-symmetric and abelian-acting line properties, it obtains a precise correspondence between orthosets and transitive Hermitian and quadratic spaces, showing that non-Boolean quadratic orthosets of rank $\ge 4$ are exactly $(P(H),\perp)$ for transitive quadratic spaces $H$, and in finite rank they are $(P(F^n),\perp)$ with $F$ a Pythagorean formally real $\star$-field. A central result is the minimality of $(P(R^n),\perp)$ for $n\ge 4$, proving it embeds into every other non-Boolean quadratic orthoset of the same rank, which identifies the Hilbert field $R$ as the least Pythagorean formally real subfield of $\mathbb{R}$ driving this hierarchy; contrastively, no such minimality holds for line-symmetric or linear orthosets. These findings connect geometric symmetry with field-embedding phenomena, offering a concrete, finite-dimensional pathway to recover real/Hilbert-space structures from combinatorial orthogonality data.

Abstract

An orthoset is a set equipped with a symmetric, irreflexive binary relation. With any (anisotropic) Hermitian space $H$, we may associate the orthoset $(P(H),\perp)$, consisting of the set of one-dimensional subspaces of $H$ and the usual orthogonality relation. $(P(H),\perp)$ 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 more and more narrow class of Hermitian spaces. We are eventually interested in quadratic spaces over countable subfields of $\mathbb R$. A line of an orthoset is the orthoclosure of two distinct element. 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 $\ell$, a group of automorphisms acting on $\ell$ 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 $n \geq 4$, the orthoset $(P(R^n),\perp)$, where $R$ is the Hilbert field, has the property of being embeddable into any other quadratic orthoset of rank $n$.

Figures (1)

• Figure 1: For a map $T \colon H_1 \to H_2$ between Hermitian spaces to be semilinear means that the diagrams on the left and in the middle commute. Here, the operations in $H_1$ and $H_2$ are marked with subscripts $1$ and $2$, respectively. For $T$ to be a semiisometry means that in addition the diagram on the right commutes.

Theorems & Definitions (71)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• Corollary 2.6
• Lemma 2.7
• proof
• Example 2.8