Transitivity and homogeneity of orthosets and inner-product spaces over subfields of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {R}}}$$\end{document}R

An orthoset (also called an orthogonality space) is a set X equipped with a symmetric and irreflexive binary relation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\perp $$\end{document}⊥, called the orthogonality relation. In quantum physics, orthosets play an elementary role. In particular, a Hilbert space gives rise to an orthoset in a canonical way and can be reconstructed from it. We investigate in this paper the question to which extent real Hilbert spaces can be characterised as orthosets possessing suitable types of symmetries. We establish that orthosets fulfilling a transitivity as well as a certain homogeneity property arise from (anisotropic) Hermitian spaces. Moreover, restricting considerations to divisible automorphisms, we narrow down the possibilities to positive definite quadratic spaces over an ordered field. We eventually show that, under the additional requirement that the action of these automorphisms is quasiprimitive, the scalar field embeds into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {R}}}$$\end{document}R.


Introduction
An orthoset is a pair (X , ⊥), where X is a set and ⊥ is a symmetric, irreflexive binary relation on X . Elements e and f such that e ⊥ f are called orthogonal and orthosets are, accordingly, also referred to as orthogonality spaces. Introduced by David Foulis and his collaborators, orthosets can be seen as an abstract version of the Hilbert space model underlying quantum physics [5,30]. Indeed, the guiding example is (P(H ), ⊥), where P(H ) is the collection of one-dimensional subspaces of a Hilbert space H and ⊥ is the usual orthogonality relation. It turns out that the orthoset (P(H ), ⊥) determines the Hilbert space H up to isomorphism uniquely. There is moreover a close correspondence between the respective automorphism groups.
In the context of the foundations of quantum mechanics, orthosets have been investigated by numerous authors, see, e.g., [3,10,13,18,25,26]. To establish conditions under which an orthoset originates from a Hilbert space, one may take advantage of the close relationship with lattice theory. Indeed, with any orthoset (X , ⊥) we may associate the complete ortholattice of orthoclosed subsets, denoted by C(X ), and the task becomes to ensure that C(X ) is isomorphic to the lattice of closed subspaces of a Hilbert space. The lattice-theoretic approach to the foundations of quantum mechanics has in turn a long tradition, going back to Birkhoff and von Neumann's seminal work [2], and led to lattice-theoretical characterisations of the Hilbert space at least in the infinite-dimensional case, see, e.g., [15,29].
Also the present paper aims at improving our understanding of the basic quantum-physical model. We take up the goal of describing the Hilbert space on the sole basis of the orthogonality relation. Lattice theory will again play a key role and we should in fact not claim that the approach adopted here differs fundamentally from previous research lines. However, we do wish to note that our point of view is in some respect uncommon. We do not view orthosets, and hence ortholattices, as the central entity around which the remaining structure is built. We rather assign to orthosets the role of underlying sets of transformation groups, the group action being required to respect orthogonality. The elements of the orthosets are not really assigned any specific meaning. What rather matters in our eyes is the notion of "change" represented by the action of a group. The orthogonality relation furthermore prescribes which actions can be combined and might be thought of as expressing the "independence of changes". Accordingly, the complex Hilbert space is not thought of as arising from a lattice of propositions, but as an entity describing "changes" in accordance with given independence demands.
These considerations have motivated us to explore, not the complex but, the projective real Hilbert space and its groups of simple rotations. The latter are meant to be the groups of rotations of some two-dimensional subspace. We consider the corresponding subgroups of the automorphism group of an orthoset and some simple conditions that we impose on these subgroups are shown to imply that the orthoset arises from an inner-product space.
We remark that, from the intuitive point of view, matters simplify considerably when dealing with the real rather than the complex case. At least in the finite-dimensional case, a real Hilbert space can conveniently be conceived as an n-sphere, opposite points being identified. This certainly helps in finding plausible conditions for its description. In particular, an n-sphere has the intuitively obvious property of allowing continuous transitions of some point to another one, leaving the points orthogonal to the starting and destination points fixed. Although complex Hilbert spaces will not be discussed here, it should be clear that the approach could be suitably extended; at least in principle, all what we need to add is the requirement that the considered symmetries are compatible with a complex structure.
Our ideas are as follows. Let O(X ) be the group of symmetries of an orthoset (X , ⊥). For each pair of distinct elements e and f of X , we consider the subgroup G e f of O(X ) that consists of the automorphisms leaving all elements orthogonal to e and f fixed. We require, first, the transitivity of (X , ⊥): some element of G e f should map e to f . We require, second, the homogeneity of (X , ⊥): any two subgroups G e f and G e f , where e = f and e = f , are conjugate via an automorphism mapping e to e . It turns out that these conditions are already sufficient to ensure that there is a Hermitian space H such that (X , ⊥) is isomorphic to (P(H ), ⊥).
In a further step, we consider, instead of the whole group G ef , the set R e f of automorphisms divisible in G e f . Requiring R e f , for any distinct e and f , to be an abelian subgroup of O(X ) and making similar assumptions as before, we again have that (X , ⊥) is representable by means of a Hermitian space. Now, however, we can say much more: we can show that the scalar division ring is a field (i.e., commutative), endowed with the identity involution, and formally real. In fact, our refined representation theorem is based on positive-definite quadratic spaces over ordered fields.
Inner-product spaces of this kind might resemble to a good extent the Hilbert spaces over R. However, it might be illusory to expect that, in the framework considered here, there are natural conditions ensuring that the scalar field of a quadratic space actually coincides with R. Our concerns relate in particular to the finite-dimensional case. However, we shall consider the following condition, fulfilled in any real Hilbert space H of dimension ≥ 3: if U is a non-trivial simple rotation of H , then the conjugates of U generate a subgroup of the orthogonal group that acts on P(H ) transitively. In other words, the group generated by the simple rotations acts quasiprimitively on P(H ); see, e.g., [23]. Adding this condition, we have that the field of scalars does not contain non-zero infinitesimals and is thus a subfield of R. In a natural sense, H then densely embeds into a real Hilbert space. We note that the quasiprimitivity of a group action is implied by its primitivity, a property that we could have employed alternatively; cf. [27].
The paper consists basically of two parts. The first part, consisting of Sects. 2-4, is largely preparatory. However, the theorem that we prove in the second half of Sect. 3 has to our knowledge not yet been stated in this way. The second part, consisting of Sects. 5-7, contains the core results, which we outlined above.
In detail, the paper is organised as follows. Section 2 is devoted to the correspondence between orthosets and ortholattices. We first shortly discuss in which way and under which conditions complete atomistic lattices may be represented by means of a suitable structure on the collection of their atoms, and we then review the case of ortholattices. In Sect. 3, we deal with the correspondence between ortholattices and linear spaces. Again, we first review the case of lattices, then the case of ortholattices. In the latter case, a well-known theorem characterises the lattice of subspaces of an (anisotropic, at least 4-dimensional) Hermitian space as a complete, irreducible, AC ortholattice. We present a modified version of this theorem, because we need a formulation that involves properties of the finite lattice elements only. Finally, it should be noted that the correspondences discussed in Sects. 2 and 3 can not easily be extended to a categorical framework. The situation is more transparent when restricting to automorphisms and in Sect. 4 we recall shortly some relevant facts.
These lengthy preparations are the basis for tackling our actual aim. In Sect. 5, we characterise Hermitian spaces by means of groups acting on orthosets. The relevant orthosets are called homogeneously transitive. We note that we generally assume the orthosets to have rank ≥ 4, but otherwise the rank is not presupposed to be finite or to be infinite. In Sect. 6, we refine our results in the sense that we are more specific about the scalar division ring. We show that so-called divisibly transitive orthosets are associated with positive-definite quadratic spaces. Finally, in Sect. 7, we consider what we call the rotation group of a divisibly transitive orthoset (X , ⊥). Under the hypothesis that this group acts quasiprimitively on X , we show that (X , ⊥) corresponds to a quadratic space over a subfield of the reals.

Atomistic lattices and their atom spaces
A central issue in this paper is the interplay between orthosets on the one hand and innerproduct spaces on the other hand. We may say that lattices act as a "mediator" between these two sorts of structures. Indeed, orthosets lead to ortholattices, and ortholattices of a certain kind are associated with inner-product spaces.
In this section, we shall compile basic definitions and facts concerning the former correspondence. In view of the needs of subsequent considerations, we adopt, however, a wider perspective: we start by discussing lattices (without an orthocomplementation) and their atom spaces, and we turn afterwards to ortholattices and their associated orthosets. For any further details, we refer the reader to Maeda and Maeda's monograph [19].

Reconstructing lattices from their atom spaces
A lattice L is called atomistic if any element is the join of atoms. For an atomistic lattice L, the collection of atoms of L will be denoted by A(L), called the atom space of L.
Let L be a complete atomistic lattice. The question seems natural whether A(L) can be equipped with a suitable structure that allows us to reconstruct L. Sending each a ∈ L to ω(a) = {p ∈ A(L) : p ≤ a}, we get an order embedding of L in the powerset of A(L). To answer our question, we should find a way of characterising the image of ω.
A possibility is to define a closure operation on A(L); see, e.g., [7].
Ordered by set-theoretic inclusion, the collection C(A(L)) of supclosed subsets of A(L) is a complete lattice: for A ι ∈ C(A(L)), ι ∈ I , we have that ι A ι is the infimum in C(A(L)) and ( ι A ι ) ∨ is the supremum. This lattice is isomorphic to L. Indeed, we easily confirm that ω establishes an isomorphism between L and C(A(L)).
In general, this observation will not help us to reduce L to a simpler structure. But we may check to which extent the finitary version of ∨ can describe L.
is called a subspace of L and we denote the complete lattice of subspaces by S(A(L)); cf., e.g., [19, (15.1)].
A complete lattice L is called compactly atomistic if L is atomistic and, for any atoms r and p ι , ι ∈ I , such that r ≤ ι∈I p ι , there is a finite subset I 0 ⊆ I such that r ≤ ι∈I 0 p ι . It is immediate that in this case, the two closure operations on A(L) coincide, that is, ∨ = − . Recall furthermore that an element a of a lattice L is finite if a is either the bottom element or the join of finitely many atoms. The set of finite elements of L is denoted by F (L). Clearly, F (L) is a join-subsemilattice of L. Lemma 1 Let L be a complete atomistic lattice.
is an order embedding preserving arbitrary meets. Restricted to the finite elements, ω establishes an isomorphism between the join-semilattices F (L) and F (S(A(L))). is an isomorphism.
Under the assumption of compact atomisticity, a lattice L can thus be described by means of its atom space A(L) equipped with the closure operator − . In the general case, we may describe in this way at least the finite part of L.
We shall now go one step further and replace the closure operator − , the finitary version of ∨ , with its binary version. For any p 1 , p 2 ∈ A(L), let p 1 p 2 = {q ∈ A(L) : q ≤ p 1 ∨ p 2 }. A set P equipped with a map : P × P → P(P) can be made into a closure space in the obvious way: for A ⊆ P we let A be the smallest superset of A such that p 1 , p 2 ∈ A implies p 1 p 2 ⊆ A . A set closed w.r.t. is called linear and the complete lattice of linear sets is denoted by L(P).
A lattice is called modular if, for each pair a and b, we have It turns out that if an atomistic lattice L is modular, then − = and hence S(A(L)) = L(A(L)), that is, the subspaces of A(L) coincide with the linear subsets of A(L) [19, (15.2)].
To characterise the operation , we are led to the following classical notion.

Definition 1
A projective space is a non-empty set P together with a map : P × P → P(P) such that, for any e, f , g, h ∈ P, the following conditions hold.
Here, we understand that is pointwise extended to subsets, that is, for e ∈ P and A ⊆ P, we put e A = {e f : f ∈ A} and similarly for A e. Conversely, let (P, ) be a projective space. Then L(P) is a compactly atomistic, modular lattice. The map P → A(L(P)), e → {e} is an isomorphism of projective spaces.

Reconstructing ortholattices from their associated orthosets
Let us contrast these familiar facts with the case that we deal with an atomistic lattice that comes equipped with an orthogonality relation. An orthocomplementation on a bounded lattice L is an order-reversing involution ⊥ that maps each element a to a complement of a. Equipped with ⊥ , L is called an ortholattice. Furthermore, L is in this case called an orthomodular lattice, or an OML for short, if a ≤ b implies that there is a c ≤ a ⊥ such that b = a ∨ c [4,17].
It turns out that to equip the atom space of a complete atomistic ortholattice with a structure determining the ortholattice is straightforward. Instead of projective spaces, we use the following notion, which is of an entirely different nature [5,30].

Definition 2
An orthoset (or an orthogonality space) is a non-empty set X equipped with a symmetric, irreflexive binary relation ⊥, called the orthogonality relation.
For a subset A of an orthoset X , we let A ⊥ = {e ∈ X : e ⊥ f for all f ∈ A} be the orthocomplement of A. The map sending any A ⊆ X to A ⊥⊥ is a closure operation and we call sets that are closed w.r.t. ⊥⊥ orthoclosed. The complete lattice of orthoclosed subsets of X is denoted C(X ) and ⊥ makes C(X ) into an ortholattice. Again, the infima in C(X ) are given by the set-theoretic intersection, and for orthoclosed subsets A ι , ι ∈ I , we have that Let L be a complete atomistic ortholattice. Elements a and b of L are called orthogonal if a ≤ b ⊥ ; we write a ⊥ b in this case. Obviously, A(L) equipped with the orthogonality relation ⊥ inherited from L is an orthoset. Moreover, we readily check that ∨ = ⊥⊥ . That is, the complete lattice of orthoclosed subsets of A(L) coincides with the complete lattice of supclosed subsets of A(L). Hence it makes sense to denote both lattices by C(A(L)) and we have that ω : L → C(A(L)), a → {p ∈ A(L) : p ≤ a} is a lattice isomorphism. We conclude that, without any additional assumptions on L, the orthogonality relation on A(L) is suitable to describe L.
We have the following correspondence between ortholattices and orthosets. We call an orthoset (X , ⊥) point-closed [25] if every singleton is closed, that is, {e} ⊥⊥ = {e} for any e ∈ X . We note that this property is equivalent to strong irredundancy [21]: for any e, f ∈ X , {e} ⊥ ⊆ { f } ⊥ implies e = f . An isomorphism between orthosets is meant to be a bijection preserving the orthogonality relation in both directions.

Proposition 2 Let L be a complete atomistic ortholattice. Then
Conversely, let (X , ⊥) be a point-closed orthoset. Then C(X ) is a complete atomistic ortholattice. The map X → A(C(X )), e → {e} is an isomorphism of orthosets.
Proof We already know that ω : L → C(A(L)) is an isomorphism of lattices. To see that ω preserves the orthocomplementation, let a ∈ L. We have ω(a) ⊥ The assertions of the second paragraph are clear.

Atomistic lattices and linear spaces
We now turn to the second afore-mentioned issue: the correspondence between lattices and linear spaces. Again, we begin by mentioning the case of linear spaces (without predefined orthogonality relation) and we discuss then in some detail the case of inner-product spaces. The basic reference is again [19]; see also [22]. In addition to preparatory material, the present section contains a representation theorem for Hermitian spaces by means of its associated ortholattice, differing from the common version in that it is based on properties of the finite part of the lattice only.

Reconstructing linear spaces from their subspace lattices
We use the shortcut sfield to refer to a skew field (i.e., to a division ring). Let H be a linear space over some sfield. We generally assume linear spaces not to have less than three dimensions but we do allow the case of infinite dimensions. We denote by L(H ) the set of subspaces of H , partially ordered by set-theoretic inclusion. Then L(H ) is a complete lattice.
Assuming a dimension ≥ 4, we may characterise H by means of L(H ). The key properties of the lattice happen to occur in Proposition 1, which describes those atomistic lattices that are determined by the relation between triples of atoms according to which the first one is below the join of the other two. There is little to add: the reducibility of the lattice as well as lengths ≤ 3 are to be excluded. For the difficult half of the subsequent fundamental theorem, see, e.g., [1,Ch. VII] or [19, (33.6)].
We call a lattice L irreducible if L is not isomorphic to the direct product of two lattices with at least two elements.

Theorem 1 Let H be a linear space. Then L(H ) is an irreducible, compactly atomistic, modular lattice.
Conversely, let L be an irreducible, compactly atomistic, modular lattice of length ≥ 4. Then there is a linear space H such that L is isomorphic to L(H ).

Reconstructing Hermitian spaces from their subspace ortholattices
We shall next see how the picture changes for linear spaces that are endowed with an orthogonality relation.
Let H be a linear space. We denote by [u 1 , . . . , u k ] the linear span of non-zero vectors Let ⊥ be a binary relation on P(H ). We call (P(H ), ⊥) an orthogeometry if (P(H ), ⊥) is an orthoset with the following properties: We may regard ⊥ in this case alternatively as a relation on H itself: . Note moreover that, by (OG1), any orthoclosed subset of P(H ) is of the form P(E) for some subspace E of H . We call E in this case orthoclosed as well and we denote the ortholattice of all orthoclosed subspaces of H by C(H ).
The notion of an orthogeometry was introduced, with a regularity assumption instead of irreflexivity, by Faure and Frölicher [9, Chapter 14.1]. A relation ⊥ making P(H ) into an orthogeometry in our sense was simply called an "orthogonality" in [8].

Lemma 2 Let H be a linear space over the sfield K and let (P(H ), ⊥) be an orthogeometry. (i) Let U be a finite-dimensional subspace of H . Then any set of mutually orthogonal nonzero vectors in U can be extended to an orthogonal basis of U . (ii) For any [u] ∈ P(H ) and any two
Proof Ad (i): Note first that, by (OG1) and the irreflexivity of ⊥, any set of mutually orthogonal vectors is linearly independent.
Assume that, for some k ≥ 1, there are pairwise orthogonal non-zero vectors u 1 , . . . , u k ∈ U and that there is As U is of finite dimension, the assertion follows.
Orthogeometries give rise to inner products. The key result originates from [2] and deals with the finite-dimensional case. For the generalisation to infinite dimensions, which we state below, see [9].
A -sfield is an sfield K equipped with an involutorial antiautomorphism : K → K . Let H be a (left) linear space over the -sfield K . By a Hermitian form on H , we mean a map (·, ·) : H × H → K such that, for any u, v, w ∈ H and α, β ∈ K , we have Endowed with a Hermitian form, we call H a Hermitian space. Note that, by the last condition which is not commonly assumed, we require a Hermitian space always to be anisotropic. If the -sfield K is commutative and the involution is the identity, we call H a quadratic space. In this case, we also assume K to be of characteristic = 2.
Let H be a Hermitian space.
is an orthogeometry, and we call ⊥ the orthogonality relation induced by (·, ·).

Theorem 2 Let H be a linear space over an sfield K and let H be of dimension ≥ 3. Let ⊥ be such that (P(H ), ⊥) is an orthogeometry. Then there is an involutorial antiautomorphism
on K and a Hermitian form (·, ·) based on such that (·, ·) induces ⊥.
We now turn to the characterisation of Hermitian spaces by the ortholattice of their closed subspaces.
A lattice L with 0 is said to fulfil the covering property if, for any a ∈ L and any atom p ∈ L such that p a, a is covered by a ∨ p. If this property holds only when a is finite, we say that L fulfils the finite covering property. An atomistic lattice with the covering property is called AC.
Moreover, the irreducibility of ortholattices is understood in the expected way: an ortholattice is irreducible if it is not isomorphic to the direct product of two ortholattices with distinct bottom and top elements. We note that the irreducibility of an ortholattice is equivalent to the irreducibility of its lattice reduct.

Theorem 3 Let H be a Hermitian space. Then
Conversely, let L be a complete, irreducible AC ortholattice of length ≥ 4. Then there is a Hermitian space H such that L is isomorphic to C(H ).
The remainder of this section is devoted to the presentation of a modified version of Theorem 3. The idea is to characterise the ortholattice by sole reference to the finite elements.
With any element u of an ortholattice L, we associate the sublattice Assume that we can make ↓u into an ortholattice whose orthogonality relation coincides with the one inherited from L. Then the orthocomplementation is Indeed, let be an orthocomplementation on ↓u such that, for a, b ≤ u, we have that a ≤ b if and only if a ⊥ b. Then, for any a ∈ ↓u, a ⊥ a implies a ≤ a ⊥ u , and a ⊥ u ⊥ a implies a ⊥ u ≤ a . We readily check that ↓u, equipped with ⊥ u , is an ortholattice if and only if ⊥ u is involutive if and only if (u ⊥ , u) is a modular pair [19, (29.10)]. Below, however, we will encounter a stronger condition, which actually ensures that ↓u is even an OML.

Lemma 3 Let L be an ortholattice and u ∈ L. Then the sublattice ↓u, endowed with ⊥ u , is an OML if and only if, for any a
Proof The "only if" part is clear by the definition of orthomodularity. To see the "if" part, assume that the latter indicated condition holds. It suffices to show that ↓u, equipped with ⊥ u , is an ortholattice. For We conclude that a ⊥ u ⊥ u = a for any a ∈ ↓u and it follows that ↓u is indeed an ortholattice.
We remark that the following lemma is shown on the basis of arguments analogous to those used for Lemma 2(i).

Lemma 4
Let L be an atomistic ortholattice such that, for any distinct atoms p and q, there is an atom r ⊥ p such that p ∨ q = p ∨ r . Then the following holds.

(i) An element a of L is finite if and only if a is the join of finitely many mutually orthogonal atoms. (ii) Let a be a finite element and p an atom of L not below a. Then there is a q ⊥ a such that a ∨ p = a ∨ q. (iii) For any finite elements a and b such that a
Proof The following claim implies both parts of the lemma: ( ) Let p 1 , . . . , p k , k ≥ 0, be mutually orthogonal atoms and let q not below their join. Then there is an atom r ⊥ p 1 , . . . , p k such that Proof of ( ): If k = 0, the assertion is trivial; assume that k ≥ 1. By assumption, there is an atom q 1 ⊥ p 1 such that p 1 ∨ q = p 1 ∨ q 1 . Similarly, there is an atom q 2 ⊥ p 2 such that Arguing repeatedly in a similar way, we see that there is an atom r = q k with the desired property.

Lemma 5 Let L be an atomistic lattice with the finite covering property. Then F (L) consists of the elements of finite height and is an ideal of L.
Proof See [19,Section 8].

Lemma 7
Let L be an atomistic ortholattice with the finite covering property and such that, for any distinct atoms p and q, there is an atom r ⊥ p such that p ∨ q = p ∨ r. Then F (L) is a modular sublattice of L.
Proof By Lemma 5, F (L) is an ideal, in particular a sublattice, of L.
Let u be a finite element of L. Then ↓u is an AC lattice of finite length. Let a ≤ b ≤ u. By Lemma 4(iii), there is a c ⊥ a such that a ∨ c = b. By Lemma 3, it follows that ↓u is an OML. By Lemma 6, ↓u is even modular.
Let a, b ∈ F (L). Then the pair a, b fulfils the modularity condition (2) in ↓(a ∨ b) and consequently also in F (L). We conclude that F (L) is modular.
We are now ready to show a version of Theorem 3 that refers to finite elements only.

Theorem 4 Let L be a complete atomistic ortholattice with the following properties:
(H1) L fulfils the finite covering property; (H2) for any distinct atoms p and q, there is an atom r ⊥ p such that p ∨ q = p ∨ r; (H3) for any orthogonal atoms p and q there is a third atom below p ∨ q; (H4) L is of height ≥ 4.

Then there is a -sfield K and a Hermitian space H over K such that L is isomorphic to the ortholattice C(H ).
Proof Recall from Sect. 2 that S(A(L)) is the complete lattice of subspaces of the atom space A(L). By Lemma 1, S(A(L)) is compactly atomistic. We claim that S(A(L)) fulfils the covering property. Indeed, let P ⊂ Q ⊆ P ∨ {p} for some P, Q ∈ S(A(L)) and p ∈ A(L). Choosing q ∈ Q \ P, we have that q ≤ r 1 ∨ . . . ∨ r k ∨ p for some r 1 , . . . , r k ∈ P, and since q ∧ (r 1 ∨ . . . ∨ r k ) = 0, it follows from (H1) that p ≤ r 1 ∨ . . . ∨ r k ∨ q and thus Q = P ∨ {p}. By Lemma 7, (H1) and (H2) imply that F (L) is a modular sublattice of L. Furthermore, by Lemma 1, ω : F (L) → F (S(A(L))), a → {p ∈ A(L) : p ≤ a} is an isomorphism of join-semilattices. As ω also preserves meets, it is actually an isomorphism of lattices. We conclude that also F (S(A(L))) is a modular sublattice of S(A(L)). By [19, (14.1)], it follows that S(A(L)) is modular.
Finally, assume that S(A(L)) is reducible. By [19, (16.6)], there are two distinct atoms below whose join there is no further atom. That is, there are distinct atoms p, q ∈ L such that there is no third atom below p ∨ q. If p and q are not orthogonal, this contradicts (H2), otherwise this contradicts (H3). We conclude that S(A(L)) is irreducible.
We summarise that S(A(L)) is an irreducible, compactly atomistic, modular lattice. As L can be order-embedded into S(A(L)), (H4) implies that the length of S(A(L)) is at least 4. By Theorem 1, there is a linear space H over an sfield K such that S(A(L)) is isomorphic to the lattice L(H ) of subspaces of H . Moreover, there is an order embedding ϕ : L → L(H ), which establishes an isomorphism between the respective sublattices F (L) and F (L(H )).
In particular, ϕ induces a bijection between the atom space A(L) and the set of onedimensional subspaces P(H ). Under this correspondence, we can make P(H ) into an orthoset. In fact, (P(H ), ⊥) is then even an orthogeometry: (OG1) obviously holds and (OG2) follows from (H2).
We conclude that, by Theorem 2, there is an involutorial antiautomorphism of K and a Hermitian form (·, ·) on H based on that induces the orthogonality relation on P(H ). As (A(L), ⊥) is isomorphic to (P(H ), ⊥), we have that C(A(L)) is isomorphic to C(P(H )). But by Proposition 2, L is isomorphic to C(A(L)), and we may naturally identify C(P(H )) with C(H ). We have hence shown that L is isomorphic to C(H ).

Automorphisms of orthosets, ortholattices, and Hermitian spaces
The ultimate goal of this paper is to characterise inner-product spaces by means of symmetries of their associated orthosets. We have discussed so far the correlation between orthosets, ortholattices, and Hermitian spaces, but we have not considered the question whether these correspondences can be extended to include the respective structure-preserving maps. We insert a short section to make up for this omission, providing some basic definitions and recalling background facts. We will focus exclusively on automorphisms.
There is an obvious correspondence between the automorphism group of a complete atomistic ortholattice and the automorphism group of its associated orthoset.
An automorphism of an orthoset (X , ⊥) is meant to be an isomorphism of X to itself, that is, a bijection ϕ : X → X such that, for any e, f ∈ X , e ⊥ f if and only if ϕ(e) ⊥ ϕ( f ). The group of automorphisms of (X , ⊥) will be denoted by O(X ). For a Hermitian space H , the correspondence between the automorphisms of (P(H ), ⊥) and those of H is described by a suitable version of Wigner's Theorem; see, e.g., [20].
A linear automorphism U of a Hermitian space H is called unitary if U preserves the Hermitian form. The group of unitary operators on H is denoted by U(H ) and we write I for its identity. We moreover denote the centre of the scalar -sfield K by Z (K ) and we let U (K ) = {ε ∈ K : εε = 1} be the set of unit elements of K .

Theorem 5 Let H be a Hermitian space over the -sfield K of dimension ≥ 3. For any unitary operator U , the map
is an automorphism of (P(H ), ⊥). The map P : Conversely, let ϕ be an automorphism of (P(H ), ⊥) and assume that there is an at least two-dimensional subspace S of H such that ϕ([x]) = [x] for any x ∈ S • . Then there is a unique unitary operator U on H such that ϕ = P(U ) and U | S is the identity.
Let (X , ⊥) be an orthoset. We shall be interested in groups of automorphisms of (X , ⊥) that act so-to-say minimally, that is, we shall deal with groups keeping everything orthogonal to some element and its image fixed. For any A ⊆ X , we define For a Hermitian space H , we may observe that O(P(H ), P(S)) can be conveniently be described by a subgroup of U(H ), provided that S ∈ C(H ) is such that S ⊥ is of dimension ≥ 2. Indeed, in this case, we let and we have, by Theorem 5, that P : U(H , S) → O(P(H ), P(S)) is an isomorphism of groups. Assume now that H is a quadratic space, that is, K is a (commutative) field and the identity. In this case, the unitary operators are commonly called orthogonal and the unitary group, now called the orthogonal group, is denoted by O(H ). For some subspace S of H , also the group defined in (4) is likewise denoted by O(H , S).
Let U ∈ O(H ) be such that, for some finite-dimensional subspace S, U | S ⊥ is the identity. Then S is an invariant subspace and U | S is an orthogonal operator on S. Furthermore, the determinant of U | S , which is 1 or −1, does not depend on S. Indeed, ifS is a further finitedimensional subspace such that U |S ⊥ is the identity, then det U |S = det U | S+S = det U | S . Hence it makes sense to define SO(H ) as the subgroup consisting of all U ∈ O(H ) such that, for some finite-dimensional subspace S of H , U | S ⊥ is the identity and U | S has determinant 1. If H is finite-dimensional, this is the usual special orthogonal group. We

Homogeneous transitivity
We now turn to the main concern of this paper: we consider orthosets that possess, in a certain sense, a rich set of symmetries and we investigate their relationship to inner-product spaces.
For any pair of distinct elements e and f of X we consider the subgroup of O(X ) Intuitively, we shall understand G e f as consisting of those automorphisms that move the element e to an element in the direction of f , in a way that the elements orthogonal to the "base point" e and the "destination point" f are kept fixed. Our guiding example is the following. We will require our orthoset (X , ⊥) to be transitive in the sense that, for any e = f , G e f actually contains an automorphism that maps e to f . Furthermore, we will postulate the homogeneity of (X , ⊥), in the sense that the subgroups G e f , e = f , of O(X ) are pairwise conjugate via an automorphism that preserves the "base points". We recall that a subgroup G 1 of O(X ) is said to be conjugate to a further subgroup

Definition 3
We call an orthoset (X , ⊥) homogeneously transitive if, for any distinct e, f ∈ X , the following holds: (HT1) There is a ϕ ∈ G e f such that ϕ(e) = f . (HT2) For any further distinct elements e and f , G e f is conjugate to G e f via an automorphism that maps e to e .
Let us check that our main example belongs to this kind of orthosets. . Hence (P(H ), ⊥) fulfils also (HT2).
As the next example shows, there are homogeneously transitive orthosets of a completely different kind.

Example 3
For any set X , (X , =) is an orthoset. We readily check that (X , =) is homogeneously transitive.
Note that, for an orthoset of the form (X , =) as indicated in Example 3, C(X ) is the Boolean algebra of all subsets of X . Hence we will call such orthosets Boolean. Boolean orthosets are certainly not what we are interested in. But we will show that the only remaining homogeneously transitive orthosets are those arising from Hermitian spaces in the same manner as indicated in Example 2.
Let us now fix a homogeneously transitive orthoset (X , ⊥) and let us assume that (X , ⊥) is not Boolean. By a ⊥-set, we mean a subset of X consisting of mutually orthogonal elements. The rank of X is the smallest cardinal number λ such that any ⊥-set is of cardinality ≤ λ. We assume X to have a rank of at least 4. Our aim is to verify that Theorem 4 applies to C(X ), the ortholattice associated with (X , ⊥).
The assertion follows.
Ad (L2): Let e and f be distinct elements of X such that e ⊥ f '. As we assume ⊥ not to coincide with =, such a pair exists. By (HT2), there is an automorphism τ such that τ (e) = e and G e f = τ −1 G e f τ . By (HT1), there is a ϕ ∈ G e f such that ϕ(e ) = f . Then τ −1 ϕτ ∈ G e f and g = τ −1 ( f ) = τ −1 ϕτ (e) ∈ G e f (e). From g ⊥ e it would follow f ⊥ e , and from g = e it would follow e = ϕ(e ) = f , both a contradiction. Hence g is an element of G e f (e) distinct from e and f . We furthermore have that G e f (e) ⊥ {e, f } ⊥ . Hence g ∈ {e, f } ⊥⊥ .
The next lemma, together with Proposition 2, implies that we may identify (X , ⊥) with the orthoset associated with C(X ).

Lemma 9 (X , ⊥) is point-closed. Consequently, C(X ) is a complete atomistic ortholattice, the atoms being {e}, e ∈ X.
Proof To show that {e} ⊥⊥ = {e} for any e ∈ X , we may argue as in case of in [28,Lemma 3.2]; the proof applies also without the assumption of a finite rank.
We now turn to the verification of conditions (H1)-(H4) of Theorem 4.
From property (L2) in Lemma 8, we conclude that below the join of two orthogonal atoms of C(X ) there is a third atom. This means that C(X ) fulfils (H3).
As we have assumed (X , ⊥) to be of rank ≥ 4, C(X ) contains at least 4 mutually orthogonal elements and hence a 5-element chain. We conclude that also (H4) holds in C(X ).
Proof Note first that, as C(X ) is an atomistic ortholattice fulfilling (H2), Lemma 4 applies.
Let A ∈ C(X ) (as a lattice element) be finite, let e / ∈ A, and let B ∈ C(X ) be such We arrive at the main result of this section.

Theorem 6 Let (X , ⊥) be a homogeneously transitive orthoset of rank ≥ 4. Then either (X , ⊥) is Boolean or else there is a Hermitian space H , possessing a unit vector in each one-dimensional subspace, such that (X , ⊥) is isomorphic to (P(H ), ⊥).
Proof Assume that (X , ⊥) is not Boolean. Then, because of Lemmas 9, 10, and 13, Theorem 4 is applicable: (X , ⊥) is isomorphic to (P(H ), ⊥) for some Hermitian space H .
By Lemma 11 and Theorem 5, there is, for any vectors u, v ∈ H • , a unitary operator mapping [u] to [v]. It follows that if there is a unit vector in H , all one-dimensional subspaces contain a unit vector. To ensure the existence of a unit vector, we "rescale" the Hermitian form if necessary; see, e.g., [15].
We note that, although we cannot say much about the scalar -sfields of Hermitian spaces that represent homogeneously transitive orthosets according to Theorem 6, it is also clear that not all -sfields are eligible.

Remark 1
Let H be an at least two-dimensional Hermitian space over a -sfield K such that each one-dimensional subspace contains a unit vector. Then K has characteristic 0; see [16] or [27,Lemma 25].
We conclude the section with some further elementary observations on homogeneously transitive orthosets that might be found interesting. In the proofs we could at some places make use of Theorem 6, but we prefer to provide direct arguments.
Given an orthoset (X , ⊥), consider the G e f -orbit of an element e ∈ X : We note next that we may formulate the conditions for orthosets to arise from Hermitian spaces in a slightly modified way. This version avoids the need of excluding the case of Boolean orthosets. Conversely, assume that the orthoset (X , ⊥) fulfils (HT1') and (HT2'). Clearly, (HT1) then holds. Furthermore, (X , ⊥) is not Boolean. Indeed, otherwise G e f would contain, for any distinct e, f ∈ X , just two elements, namely, the identity and the map interchanging e and f , whereas (HT1') implies the existence of a third map.
Let Finally, we provide a further reformulation, which emphasises to some extent the role of the orbits. We might observe in this case a resemblance with the axioms of projective geometry in Definition 1. Indeed, condition (HT1") can be considered as similar to the requirement (PS1), according to which every line contains the two points by which it is spanned. Furthermore, condition (HT2") may be seen as a weakened form of (PS2), according to which any two points lie on a unique line. Remarkably, (PS3) or some other variant of the Pasch axiom does not occur. Proof Let (X , ⊥) be homogeneously transitive and not Boolean. Again, for e = f , Lemma 8 implies that {e, f } ⊥⊥ contains a third elements as well as an element orthogonal to e. Hence (HT1") follows from Lemma 14. Furthermore, by Lemma 13, C(X ) has the finite covering property. Hence (HT2") follows from Lemma 14 as well.

Divisible transitivity
We any pair e, f of distinct elements of an orthoset we have associated the group G e f of automorphisms that keep the elements orthogonal to e and f fixed. Intuitively, we have viewed G e f as describing transitions from e into the direction determined by f . The present section is based on the idea to add the requirement that these transitions may proceed in a continuous manner. We will, however, not deal with topologies, we rather propose a divisibility condition.
For distinct elements e and f of an orthoset (X , ⊥), let us consider the following collection of automorphisms: such that ψ(x) = x for all x ⊥ e, f and ψ k = ϕ}.
We shall require R e f to be a subgroup, subjected to similar conditions as G e f in case of homogeneous transitivity. Our above intuitive ideas furthermore motivate us to require R e f to be abelian: assuming that ϕ 0 ∈ R e f is an automorphism mapping e to f and, for any i ≥ 0, ϕ i+1 is a square root of ϕ i in R e f , we intend to view the subgroup generated by these maps, which is abelian, as "dense" in R e f .

Definition 4
We call an orthoset (X , ⊥) divisibly transitive if, for any distinct e, f ∈ X , the following holds: (DT0) R e f is an abelian subgroup of O(X ).
(DT1) There is a ϕ ∈ R e f such that ϕ(e) = f . We have the following, provisional representation for divisibly transitive orthosets.

Lemma 15
Let (X , ⊥) be a divisibly transitive orthoset of rank ≥ 4. Then there is Hermitian space H over some -sfield K , possessing a unit vector in each one-dimensional subspace, such that (X , ⊥) is isomorphic to (P(H ), ⊥).
Proof Note first that (X , ⊥) cannot be Boolean. Indeed, in this case R e f , for any e = f , would consist of the identity alone, in contradiction to (DT1). Lemmas 8-13 and hence Theorem 6 hold also for divisibly transitive orthosets. To show this, we argue on the basis of (DT1) and (DT2) instead of (HT1) and (HT2), respectively.
We note that Lemma 14 possesses also a version for divisibly transitive orthosets. Proof Again, we may follow literally the proof of Lemma 14, which contains the analogous statement on homogeneously transitive orthosets.
We furthermore remark that we may formulate the axioms of divisible transitivity in a slightly different way. Furthermore, we will say that a subgroup G of U(H ) acts transitively on P(S) if, for any Lemma 17 Let u, v ∈ H be linearly independent.
Proof By Theorem 5, P : consists of the divisible elements of G [u][v] and hence of the maps P(U ) such that U ∈ U(H , [u, v]) and for each k ≥ 1 there is some V ∈ U(H , [u, v] [v] . It follows that R(H , [u, v]) is an abelian subgroup of U(H , [u, v]) and also P :

Lemma 18 H is a quadratic space.
Proof We have to show that the involution on K is the identity. It will then follow that K is commutative and the lemma will be proved.
Let We proceed by showing a sequence of auxiliary statements. We will frequently use the fact that, by Lemma 17, R(H , F) is an abelian subgroup of U(H , F) acting transitively on P(F).
From the proof of the previous lemma, it is obvious how to characterise the groups R [u][v] , where u, v ∈ H are linearly independent. We insert the following lemma for later use. for some α, β ∈ K such that α 2 + β 2 = 1.
We shall finally establish that K is a formally real field. For further information on ordering on fields we refer the reader to [24, §1].
Note that, by the next lemma, K is a Pythagorean field.
Lemma 21 K is formally real. K being equipped with any order, the Hermitian form on H is positive definite.
Moreover, assume K to be equipped with an order and let u ∈ H • . Let v ∈ H be a unit vector in [u]. Then we have u = αv for some α ∈ K \ {0} and it follows (u, u) = α 2 > 0.
We summarise what we have shown.

Theorem 7
Let (X , ⊥) be a divisibly transitive orthoset of rank ≥ 4. Then there is an ordered field K and a positive-definite quadratic space H over K , possessing a unit vector in each one-dimensional subspace, such that (X , ⊥) is isomorphic to (P(H ), ⊥).

Quasiprimitive orthosets
The orthoset arising from a real Hilbert space of dimension ≥ 4 is divisibly transitive and we have seen that any divisibly transitive orthoset of rank ≥ 4 arises from a positive-definite quadratic space H over an ordered field K . We certainly wonder under which additional, natural conditions on the orthoset (P(H ), ⊥), K is actually the field of real numbers. A promising way to approach this question, however, seems hard to define and we are actually not convinced that a solution is feasible in the present framework. We will discuss here the related problem of finding reasonable conditions under which K is a subfield of R. We consider to this end a property of orthosets that is once more related to transitivity.
Let (X , ⊥) be a divisibly transitive orthoset. We refer to an automorphism contained in a subgroup R e f , where e and f are distinct elements of X , as a simple rotation. The identity map is called the trivial rotation. The subgroup of O(X ) generated by all simple rotations will be called the rotation group of (X , ⊥), denoted by R(X ).

Definition 5
We call an orthoset (X , ⊥) quasiprimitive if, for any non-trivial simple rotation , the normal subgroup of R(X ) generated by acts transitively on X .
That is, we call an orthoset quasiprimitive if the transformation group R(X ) has this property; see, e.g., [23]. More explicitly, let τ = τ −1 τ be the conjugate of some automorphism via a further automorphism τ . The quasiprimitivity of (X , ⊥) means that, given any simple rotation = id and any two points e, f ∈ X , there are τ 1 , . . . , τ k ∈ R(X ) and We claim that (P(H ), ⊥) is quasiprimitive. Let = id be a simple rotation of (P(H ), ⊥) and let [u], [v] ∈ P(H ) be distinct. Let S be a 3-dimensional subspace of H such that u, v ∈ S. SO(S), the special orthogonal group of S, is simple; see, e.g., [12]. Hence the conjugates of any U ∈ SO(S), distinct from the identity, generate the whole group SO(S) and since SO(S) acts transitively on P(S), some finite product of conjugates of U maps [u] to [v]. We conclude that some finite product of conjugates of maps [u] to [v].
Let us now fix a positive-definite quadratic space H over an ordered field K such that each one-dimensional subspace contains a unit vector and assume that the orthoset (P(H ), ⊥) is divisibly transitive.
We may describe the rotation group R(P(H )) as follows. The following definitions and facts are due to Holland [14], for further details see also [27]. We call an element α ∈ K infinitesimal if |α| < 1 n for all n ∈ N \ {0}, and we call α ∈ K finite if |α| < n for some n ∈ N \ {0}. We denote the set of infinitesimal and finite elements by I K and F K , respectively. I K and F K are additive subgroups of K and are closed under multiplication. Furthermore, F K \ I K is a multiplicative subgroup of K \ {0}, and we have I K · F K = I K and F K + I K = F K .
Likewise, a vector x ∈ H is called infinitesimal if so is (x, x), and x is called finite if so is (x, x). The sets of infinitesimal and finite vectors are denoted by I H and F H , respectively. I H and F H are subgroups of H and we have I K · F H = F K · I H = I H , F K · F H = F H , and F H + I H = F H . Furthermore, (x, y) ∈ I K if x, y ∈ F H and at least one of x and y is infinitesimal.
If the only infinitesimal element of K is 0, K is called Archimedean. In this case, K is isomorphic to a subfield of R equipped with the inherited natural order; see, e.g., [11].
For Proof Ad (i): ≈ is clearly reflexive and symmetric. To see that ≈ is also transitive, let x, y, y , z ∈ F H \ I H such that x − y, y − z ∈ I H and y = α y for some non-zero α ∈ K . Then α 2 = y , y (y, y) −1 ∈ F K \ I K and hence αz ∈ F H \ I H . We conclude that x − αz ∈ I H . If K is Archimedean, x ∈ I H implies (x, x) = 0 and hence x = 0. Hence ≈ is the equality. Conversely, if K is not Archimedean, let α ∈ I K \ {0} and let u, v ∈ H be orthogonal unit vectors.

Lemma 23
Assume that K is non-Archimedean. Then there is a U ∈ SO(H ) such that P(U ) is distinct from the identity and U (x) − x is infinitesimal for any finite vector x.

Lemma 24 If (P(H ), ⊥) is quasiprimitive, then K is Archimedean and hence a subfield of R.
Proof Let us assume that (P(H ), ⊥) is quasiprimitive and K is not Archimedean. In accordance with Lemma 23, choose some U ∈ SO(H ) such that P(U ) is distinct from the identity and U (x) − x ∈ I H for any x ∈ F H . Then we have that U ([x])≈[x] for any x ∈ H • .
Furthermore, for any further orthogonal operator V , V −1 U V has the same properties. Indeed, for any x ∈ F H , we have for any x ∈ H • .
We conclude that the orbit of any [x] ∈ P(H ) under the action of conjugates of U is contained in the ≈-class of [x]. The latter is, by Lemma 22(ii), a proper subset of P(H ). In view of Proposition 7, it follows that (X , ⊥) is not quasiprimitive.
We may summarise the results of this section as follows.
Theorem 8 Let (X , ⊥) be an orthoset of rank ≥ 4. Assume that (X , ⊥) is divisibly transitive and quasiprimitive. Then there is a positive-definite quadratic space H over a subfield of R, possessing a unit vector in each one-dimensional subspace, such that (X , ⊥) is isomorphic to (P(H ), ⊥).
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