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

An orthoset is a set equipped with a symmetric, irreflexive binary relation. With any (anisotropic) Hermitian space <i>H</i>, we may associate the orthoset <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mo>⊥</mml:mo> <mml:mo>)</mml:mo></mml:mrow> </mml:math> , consisting of the set of one-dimensional subspaces of <i>H</i> and the usual orthogonality relation. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mo>⊥</mml:mo> <mml:mo>)</mml:mo></mml:mrow> </mml:math> determines <i>H</i> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> .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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>4</mml:mn></mml:mrow> </mml:math> , the orthoset <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>P</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:msup><mml:mi>R</mml:mi> <mml:mi>n</mml:mi></mml:msup> <mml:mo>)</mml:mo></mml:mrow> <mml:mo>,</mml:mo> <mml:mo>⊥</mml:mo> <mml:mo>)</mml:mo></mml:mrow> </mml:math> , where <i>R</i> is the Hilbert field, has the property of being embeddable into any other quadratic orthoset of rank <i>n</i>.