Bands in partially ordered vector spaces with order unit

In an Archimedean directed partially ordered vector space $X$ one can define the concept of a band in terms of disjointness. Bands can be studied by using a vector lattice cover $Y$ of $X$. If $X$ has an order unit, $Y$ can be represented as $C(\Omega)$, where $\Omega$ is a compact Hausdorff space. We characterize bands in $X$, and their disjoint complements, in terms of subsets of $\Omega$. We also analyze two methods to extend bands in $X$ to $C(\Omega)$ and show how the carriers of a band and its extensions are related. We use the results to show that in each $n$-dimensional partially ordered vector space with a closed generating cone, the number of bands is bounded by $\frac{1}{4}2^{2^n}$ for $n\geq 2$. We also construct examples of $(n+1)$-dimensional partially ordered vector spaces with ${2n\choose n}+2$ bands. This shows that there are $n$-dimensional partially ordered vector spaces that have more bands than an $n$-dimensional Archimedean vector lattice when $n\geq 4$.


Introduction
Bands play an important role in the theory of vector lattices. Together with concepts such as ideals and disjointness they are widely used to study the structure of vector lattices. Bands and disjointness are closely related, as in Archimedean vector lattices bands are exactly those subspaces that coincide with their double disjoint complements. In the study of operators on vector lattices there is a particular interest in operators preserving certain structures of the lattice. Especially the study of band preserving and disjointness preserving operators is an active field of research, see e.g. [1,2,3,11].
If one considers vector lattices, one naturally encounters ordered spaces that are not lattices. For instance, linear subspaces of vector lattices and spaces of operators between vector lattices are partially ordered vector spaces, but not lattices, in general. It is therefore of interest to extend the theory of vector lattices and operators on them to partially ordered vector spaces.
The notion of a band is defined in partially ordered vector spaces by means of disjointness in [5]. Fundamental properties of bands in partially ordered vector spaces are shown in [6] with the aid of vector lattice covers.
In [9] a functional representation method provides a vector lattice cover of an Archimedean partially ordered vector space X with unit by embedding X into a space of continuous functions C(Ω), where Ω is a compact Hausdorff space. This raises the question whether the bands in X can be identified in terms of subsets of Ω. In Section 2 we establish a one-to-one correspondence between the bands in X and so-called 'bisaturated' subsets of Ω.
It has been shown in [6] that bands in X can be extended to bands in a vector lattice cover of X. In Section 3 we consider two different extension methods using disjoint complements, for which we establish the relations between the 'carrier' of a band in X and the carrier of the extension of the band in C(Ω).
In Section 4 we use the results of Section 2 to investigate bands in finite dimensional spaces. We show that the results provide a method to list all bands in spaces with polyhedral cones. Further, we prove that the number of bands in a space with a polyhedral cone is bounded by a constant only depending on the dimension of the space. By a plastering approximation we extend this result to arbitrary generating closed cones.
In the remainder of this section we fix our notation. Let X be a real vector space and let K be a cone in X, i.e., K is a wedge (x, y ∈ K, λ, µ ≥ 0 imply λx + µy ∈ K) and K ∩ (−K) = {0}. In X a partial order is introduced by defining y ≥ x if and only if y − x ∈ K. In this case X is called a partially ordered vector space. The space X is called directed if X = K − K; in this case K is called generating. The space X is Archimedean if for any x, y ∈ X with nx ≤ y for all n ∈ N we have x ≤ 0.
For a subset M of X the set of all upper bounds of M is denoted by M u . As in [4, Definition 2.1], we call M ⊆ X solid if from x ∈ X, y ∈ M and {x, −x} u ⊇ {y, −y} u it follows that x ∈ M . A solid linear subspace of X is called an ideal. According to [5], x, y ∈ X are called disjoint, in symbols x ⊥ y, if {x + y, x − y} u = {x − y, −x + y} u .
The disjoint complement of a subset M of X is the set If X is an Archimedean vector lattice, these notions coincide with the usual ones. In a directed Archimedean partially ordered vector space X every band is an ideal [6,Theorem 5.14], and for a subset B ⊆ X the set B d is a band [5,Proposition 5.5 (ii)]. Throughout the exposition we assume that X is an Archimedean partially ordered vector space and has an order unit u ∈ K, i.e., for each x ∈ X there exists α ≥ 0 such that Then X is directed. As usual, an order unit u ∈ K gives rise to a norm · u on X by With respect to this norm each positive linear functional ϕ : X → R is continuous, as |ϕ(x)| ≤ ϕ(u) for all x ∈ X with x u ≤ 1. In the norm dual X ′ the set is a cone. Next we recall the functional representation of X. Denote By the Banach-Alaoglu theorem the closed unit ball B ′ of X ′ is w*-compact. Σ is a w*-closed subset of B ′ , and hence w*-compact. Let Λ := {ϕ ∈ Σ : ϕ is an extreme point of Σ}.
Recall that ϕ ∈ Σ is extreme if ϕ is not in the (relative) interior of a linesegment in Σ, and that Λ = ∅ by the Krein-Milman theorem. In general, Λ need not be w*-closed, not even if X is finite dimensional. For M ⊆ X ′ the w*-closure of M is denoted by M . Thus Λ is a compact Hausdorff space.
In the sequel we write Φ : X → C(Λ) to denote the linear map (Φ, Λ) is a functional representation of X, which means that Φ is linear, bipositive, maps u to the constant 1 function (which implies isometry with respect to the u-norm) and the image of Φ separates the points of Λ. Recall that a linear map Ψ : U → V , where U and V are partially ordered vector spaces, is called bipositive if for every x ∈ U one has x ≥ 0 if and only if Ψ(x) ≥ 0. We proceed by recalling the vector lattice cover. A linear subspace D of a partially ordered vector space Y is order dense if every h ∈ Y is the greatest lower bound of the set {g ∈ D : g ≥ h}. A vector lattice cover of a partially ordered vector space X is a vector lattice Y for which there exists a bipositive linear map i : X → Y such that i(X) is order dense in Y . If in addition there is no proper vector sublattice of Y containing i(X), then (Y, i) is called a Riesz completion [7]. A Riesz completion Y is unique up to isomorphism [7,Remark 3.2], and hence we may speak of the Riesz completion of X. The partially ordered vector spaces that have a Riesz completion are called pre-Riesz spaces. It is shown in [7, Theorems 1.7, 3.5, 3.7, 4.13 and Corollaries 4.9-4.11] that, among other spaces, every directed Archimedean partially ordered vector space X has a Riesz completion. If, in addition, there is an order unit in X, the Riesz completion can be obtained by means of Φ. Theorem 1. [9, Theorem 10] Let X be an Archimedean partially ordered vector space with order unit u. Then C(Λ) is a vector lattice cover of X. Moreover, the Riesz subspace of C(Λ) generated by Φ(X) is the Riesz completion of X.
If (Y, i) is a vector lattice cover of X, then x ⊥ y if and only if i(x) ⊥ i(y) [5,Theorem 4.1]. In this setting, the extension property for bands is shown in [6,Proposition 5.12], namely, if B is a band in X, then there is a band B in Y such that i(B) = B ∩ i(X). To describe B by means of the functional representation, we first recall well-known results concerning ideals and bands in C(Ω), where Ω is a compact Hausdorff space, cf. e.g. [12,Example 9.4]. For a set S ⊆ C(Ω) the carrier of S is defined by  We combine properties of disjointness in vector lattice covers and in C(Ω). If X is an Archimedean partially ordered vector space with unit and Φ is its embedding into C(Λ) as given above, then Theorem 1 yields that for x, y ∈ X ⇐⇒ for each ϕ ∈ Λ one has ϕ(x) = 0 or ϕ(y) = 0.
Moreover, if B is a band in X, then there exists a regularly open set O ⊆ Λ such that B = {x ∈ X : ϕ(x) = 0 for all ϕ ∈ Λ \ O}.

Characterization of bands
We consider an Archimedean partially ordered vector space X with order unit and its embedding Φ into C(Λ) as given in Section 1. For B ⊆ X we will deal with car(Φ(B)), which we abbreviate by car(B).
where Ω is a compact Hausdorff space, then Λ = Λ and Ω are homeomorphic, such that the carrier of B ⊆ C(Ω) defined as a subset of Λ corresponds to the carrier of B seen as a subset of Ω. In that sense the definition of the carrier in X is compatible with the definition of the carrier in Section 1.
We denote For a set M ⊆ Λ we also use the notation M c := Λ \ M . span (M ) denotes the linear subspace of X ′ spanned by M and the affine hull of M is given by For M ⊆ Λ denote the zero set induced by M by In the present section, we will characterize those sets M ⊆ Λ that belong to a band B in X in the sense that B = Z(M ) and M = N (B). We need the following notion. Instead of the linear span one could also use the affine hull in the definition of saturation, see Corollary 7 below.
It is straightforward that for B ⊆ X the set N (B) is saturated. In the subsequent proposition we characterize the saturation by means of zero sets. Let us first state two properties.   Proof. If ϕ ∈ Λ ∩ span (M ) and x ∈ Z(M ), then ϕ(x) = 0, so Since the right hand side of the equality is a w*-closed set, it follows that By the Hahn-Banach theorem, there is x ∈ X such that ψ(x) = 0 and for all ϕ ∈ span (M ) one has ϕ(x) = 0. In particular, x ∈ Z(M ). So, It can be observed that the saturation of M ⊆ Λ is the largest subset N of Λ such that Z(N ) = Z(M ).
In the considerations below it appears more natural to use the affine hull instead of the linear span in describing saturations. It is a consequence of Proposition 6 that the same sets are obtained.
As a preparation for the main result concerning the characterization of bands by means of subsets of Λ, we provide some lemmas.
As a direct consequence of Lemma 9, we obtain the following result. Therefore, in analogy to bands in C(Ω) (which were discussed in Section 1), a band in X is determined whenever its carrier is known.
The next definition introduces those saturated subsets of Λ the zero sets of which turn out to be bands. Just as a band and its disjoint complement, also such saturated subsets appear in pairs. The ensuing definition makes sense in the more general setting of a topological vector space E (instead of X ′ ) and a subset V of E (instead of Λ). With hindsight and in view of We claim that the set N (B) = 1 2 , 1 is bisaturated. Indeed, 1 2 is not saturated, since every saturated set is closed.
The next lemma relates bisaturated sets and bisaturated pairs in V , where V is a subset of a topological vector space E, and provides a geometric description of bisaturated sets. Observe that for a bisaturated pair Lemma 14. For two saturated sets M 1 , M 2 ⊆ V the following four statements are equivalent.
For E = X ′ and V = Λ, we relate bisaturated pairs in Λ to disjoint complements in X.
Proposition 15. Let M 1 , M 2 ⊆ Λ be saturated sets and B 1 = Z(M 1 ) and Then for every y ∈ B 1 we have x ⊥ y and hence Φ(x) ⊥ Φ(y). Therefore, Φ(x)(ϕ) = 0 for every Again by Proposition 6, We arrive at the main result in this section. The assumption in the theorem is natural due to Proposition 10. In Section 4 we will use Theorem 16 to bound the number of bands in finite dimensional partially ordered vector spaces.

Extension of bands
Let X be a pre-Riesz space and let (Y, i) be a vector lattice cover of X, for instance its Riesz completion.
If B is a band in X, then there are two natural procedures to construct a band B in Y such that They are as follows: In [6, Proof of Proposition 5.12] it is shown that (3) is satisfied.
In this case, the subsequent statement shows (3).  We proceed by discussing procedure (II). First we show that the carrier of B and the carrier of B have the same closure (in other words, the supports of B and B are the same).
Proposition 21. For a band B in X one has car(B) = car(Φ(B) dd ).
Observe that every saturated subset of Λ is w*-closed. On the contrary, a w*-closed subset of Λ is not saturated, in general (note that in Example 36 the set {f 1 , f 2 , f 3 } is w*-closed but not saturated). Nevertheless, a statement similar to Proposition 21 is true, concerning the saturations of car(B) and car(Φ(B) dd ). Proof. Due to Theorem 16, (car(B)) c is bisaturated. By Proposition 24 we obtain (car(B)) c = sat ((sat (car(B))) c ) = sat ((sat (car(Φ(B) dd ))) c ), which implies the assertion.
Summing up the above results, the extension method (II) is such that car(B) and car(Φ(B) dd ) have the same closure and the same saturation, and if one of the two sets is known then the other one is determined.
Where procedure (II) yields the smallest band B in Y extending a band B in X, there is no largest one, in general, as the next example shows.
Example 26. Let S be the unit circle in R 2 and let X be the space of restrictions to S of affine functions from R 2 to R with pointwise order on S. C(S) is a vector lattice cover of X

Bands in finite dimensional spaces
In this section we shall see how to use the characterization of bands in terms of bisaturated sets of Λ, given in Theorem 16, to analyze the following question.
Question 27. Does there exist an a-priori upper bound for the number of bands in a finite dimensional partially ordered vector space X with a closed and generating cone K in terms of the dimension of X? If so, what is the optimal upper bound?
Clearly the number of bands in n-dimensional partially ordered vector space is 2 if n = 1. For n ≥ 2 we have the following result.
Theorem 28. The number of bands in an n-dimensional partially ordered vector space (X, K), where is K is a closed generating cone and n ≥ 2, does not exceed 1 4 2 2 n . The proof of this result consists of two parts. First it will be shown that the upper bound holds for partially ordered vector spaces with polyhedral cones. Subsequently we use the Plastering Lemma 34 to show that the same upper bound holds for general cones.
Let X be a finite dimensional vector space. For a subset M ⊆ X we mean by dim M the dimension of aff M . Recall that a closed cone K in X is polyhedral if it is the intersection of finitely many closed half-spaces. We use basic results in polyhedral geometry, which can be found, e.g., in [13]. If K is a closed polyhedral cone in X with non-empty interior, then each u in the interior of K is an order unit. Fix such a u. Then there exist m ≥ dim X functionals, ϕ 1 , . . . , ϕ m ∈ X ′ with ϕ i (u) = 1 for all i, so that The functionals ϕ i define the facets of K. Recall that a facet of a polyhedral cone K is an (exposed) face for all i}, then we can use the Hahn-Banach separation theorem to find z ∈ X ′′ = X such that ψ(z) < 0 and ϕ i (z) ≥ 0 for all i. As K = {x ∈ X : ϕ i (x) ≥ 0 for all i}, we see that z ∈ K, which is impossible, as ψ ∈ K ′ and ψ(z) < 0.
Let Λ be the set of extreme points of Σ := {ϕ ∈ K ′ : ϕ(u) = 1}. We claim that Λ = {ϕ 1 , . . . , ϕ m }, which is a closed set. Indeed, if there exist σ and ρ in Σ such that λσ + (1 − λ)ρ = ϕ i for 0 < λ < 1 and some i, then F = {x ∈ K : ϕ i (x) = 0} is contained in ker σ ∩ker ρ. As dim F = dim X −1, we find that σ = αϕ i and ρ = βϕ i for some α, β ∈ R. It now follows that The space X with polyhedral cone K as defined above fits in the setting of the functional representation given in Section 1. As K is closed, X is Archimedean. By Theorem 1, Next we state our main result on the number of bands in spaces with polyhedral cones.
Theorem 30. The number of bands in an n-dimensional partially ordered vector space X ordered by a generating polyhedral cone K does not exceed 1 4 2 2 n , provided n ≥ 2.
By using Theorem 16, the proof of Theorem 30 reduces to a purely combinatorial problem concerning polytopes. Indeed, note that in the setting above, the set Σ is a closed convex polytope in the (n − 1)-dimensional affine hyperplane H = {ϕ ∈ X ′ : ϕ(u) = 1} with vertex set V = {ϕ 1 , . . . , ϕ m }. From Theorem 16 we know that B is a band in X if and only if N (B) is a bisaturated set in Λ. Therefore the number of bands equals the number of bisaturated sets.
Given a finite set V in X, Definition 12 entails that a set S ⊆ V is bisaturated in V if and only if and {S, T } is a bisaturated pair in V if and only if Thus, to prove Theorem 30 it suffices to show the following combinatorial statement.
Theorem 31. In a vertex set V of a polytope P in R n−1 , with n ≥ 2, there are at most 1 4 2 2 n bisaturated sets. Proof. Suppose that there exists a polytope P in R n−1 with q bisaturated sets in the vertex set V of P . We wish to show that q ≤ 1 4 2 2 n . Without loss of generality we may assume that P is chosen such that the number of its vertices is minimal among the cardinalities of the vertex sets of polytopes in R n−1 with at least q bisaturated sets.
Assuming the claim for the moment, we see that V has at most 2 n − 2 points, so the number of bisaturated sets of P cannot exceed 2 2 n −2 . So, it remains to prove the claim. Obviously, the claim is true for k = 1, as no three vertices of P lie on a line.
For the sake of contradiction, assume that there exists a k-dimensional affine subspace U , with k ≥ 2, such that Now take k ≥ 2 as small as possible.
For a proof of Claim 2, observe that U ⊆ aff W implies that U ′ := U ∩ aff W has dimension l < k and |U ′ ∩ V | > 2 k − 2 ≥ 2 l+1 − 2, which would mean that k is not minimal.
We continue the proof of Claim 1 and fix u ∈ V ∩ U . To arrive at a contradiction we show that the polytope P ′ with vertex set V ′ = V \ {u} has the property that if S is a bisaturated set in V , then S ′ = S \ {u} is a bisaturated set in V ′ . Moreover, V ′ contains at least as many bisaturated sets as V . The last statement contradicts the minimality assumption on the number of vertices of P .
Without loss of generality we may assume that |S ∩ U | > 2 k − 1. Then U ∩ V ⊆ S, as U ⊆ aff S (by Claim 2) and V ∩ aff S = S, and hence u ∈ S. This implies that V \ S = V ′ \ S ′ ; so, To show that S ′ = V ′ ∩ aff (V ′ \ T ′ ) we distinguish two cases: u ∈ T and u ∈ T . If u ∈ T , then We conclude that {S ′ , T ′ } is a bisaturated pair of P ′ .
It remains to show that P ′ has at least as many bisaturated sets as P . Recall that for each bisaturated set S of P , S ′ = S \ {u} is a bisaturated set of P ′ . So, it suffices to show that if S is a bisaturated set of P and u ∈ S, then S ′ is not a bisaturated set of P . Assume that both S and S ′ are bisaturated sets of P . Note that if |S ∩ U | > 2 k − 1, then |S ′ ∩ U | > 2 k − 2. But this implies (by Claim 2) that U ⊆ aff (S ′ ), so V ∩ U ⊆ V ∩ aff S ′ = S ′ , which is impossible, as u ∈ S ′ . Now assume that |S ∩ U | ≤ 2 k − 1; so |(V \ S) ∩ U | ≥ 2 k > 2 k − 1 and then (by Claim 2 for the last time) U ⊆ aff (V \ S). Let T = V ∩ aff (V \ S) and R = V ∩ aff (V \ S ′ ). Obviously, if we can show that T = R, then it follows that S = S ′ and we are done. Note that, as u ∈ aff (V \ S), aff (V \ S) = aff (V \ S ′ ), and hence T = R.
We next generalize the upper bound in Theorem 30 for the polyhedral case to the general finite dimensional case. Let X be a finite dimensional vector space and K a generating closed cone in X. We denote by (X, K) the partially ordered vector space X ordered by K. We fix an interior point u of K and consider the corresponding functional representation (Φ, Λ) as in Section 1. Given bands B 1 , B 2 , . . . , B q in (X, K), the idea now is to construct a polyhedral cone L in X containing K such that each B i is a band in (X, L). The construction uses the characterisation of bands by means of bisaturated sets of Λ and the following two technical lemmas.
If {M 1 , M 2 } is a bisaturated pair in Λ and we replace M 1 and M 2 by smaller closed sets W 1 and W 2 that have the same affine hulls as M 1 and M 2 , respectively, then the next lemma says that the pair of their saturations in W = W 1 ∪ W 2 is a bisaturated pair in W .
and if F is a closed subset of Λ with W 1 ∪ W 2 ⊆ F , then for the sets Similarly, F ∩ aff (F \ N 2 ) = N 1 .
If ϕ ∈ Λ and x ∈ B are such that ϕ(x) = 0, then there are (ϕ α ) α in Λ with ϕ α → ϕ (in weak* topology) and then ϕ α (x) = 0 for α large. Therefore ϕ α ∈ C for α large, so ϕ ∈ C. Thus aff (car(B)) ⊆ aff (C) = aff (C). Now we are in a position to present the plastering lemma. Choose, with the aid of Lemma 33, finite sets such that aff (U i ) = aff (car(B i )) and aff (V i ) = aff (car(B i d )). Define and L := {x ∈ X : ϕ(x) ≥ 0 for every ϕ ∈ F }. Since for each i, and, similarly, due to (2). Hence Λ ⊆ aff (F ), so that L is a cone. Also, since F ⊆ Λ, we have L ⊇ K, so that L is generating. Moreover, the order unit u of (X, K) is an interior point of K, hence an interior point of L and therefore an order unit in (X, L).
Every element of F is an extreme point of Indeed, if ϕ ∈ F , 0 < λ < 1, and ϕ 1 , ϕ 2 ∈ Σ L are such that ϕ = λϕ 1 + (1 − λ)ϕ 2 , then ϕ ∈ Λ and ϕ 1 , ϕ 2 ∈ Σ, so ϕ 1 = ϕ 2 = ϕ, as Λ consists of extreme points of Σ. In conclusion, L is a generating polyhedral cone in X, u is an order unit in (X, L), and F is the set of extreme points of Σ L . It remains to show that for each i the subspace B i is a band in (X, L).
According to Lemma 32, the saturations in F of the sets U i and V i constitute a bisaturated pair in F , so that, according to Proposition 15, Z(U i ) and A combination of Theorem 30 and Lemma 34 yields Theorem 28. We believe that the upper bound in Theorem 28 is far from optimal. The following example shows that there exist (n + 1)-dimensional partially ordered vector spaces with 2n n + 2 bands. In particular, it follows that there exist partially ordered vector spaces with dimension n ≥ 4 that have more bands than an n-dimensional Archimedean vector lattice, which has 2 n bands.
Example 35. Let P be a polytope in R n with vertex set V = {v 1 , . . . , v 2n } that is in general linear position, i.e., for each affine subspace U ⊂ R n with dim U < n we have that |V ∩ U | ≤ dim U + 1. In other words, there are no 3 points in V on a line, no 4 points in V in a plane, etc. To show existence of such a set V , let n ≥ 1 and S n−1 be the unit sphere in R n . Choose uniformly at random v 1 , . . . , v 2n from S n−1 . Then the convex hull of {v 1 , . . . , v 2n } is, with probability 1, a polytope, P , with non-empty interior. If necessary apply a translation, so that 0 is in the interior of P . By construction, the vertex set V = {v 1 , . . . , v 2n } of P is in general linear position.
Next we prove that (R n+1 , K V ) has 2n n +2 bands by showing that S ⊆ Λ is bisaturated if and only if S = ∅, S = Λ, or |S| = n. Clearly, if S is a bisaturated set in Λ and |S| > n, then dim S > n − 1, as V is in general linear position. So, S = Λ ∩ aff S = Λ. Likewise, if S is bisaturated in Λ and |S| < n, then |Λ \ S| > n, and hence S = ∅.
Observe that span {ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 } has dimension 3. Indeed, ϕ 1 + ϕ 2 − ϕ 3 − ϕ 4 = 0, and ϕ 1 , ϕ 2 , ϕ 3 are linearly independent. We embed (R 4 , K) into (R 5 , R 5 + ) under Φ. We want to determine all bands in (R 4 , K). Recall that a band B in a partially ordered vector space is directed if for each x, y ∈ B there exists z ∈ B such that x ≤ z and y ≤ z. In the examples above the partially ordered vector spaces have directed and non-directed bands, whereas in vector lattices all bands are directed. The next example shows that there exists a partially ordered vector space whose non-trivial bands are all non-directed. Here it turns out that the carriers of B in the extension procedures (I) and (II) (see Section 3) differ, in contrast to the situation in Example 18.
Example 37. Let P be the polytope, with vertex set v 1 , . . . , v 9 , depicted in Figure 1. It is interesting to analyze the number of directed bands in an n-dimensional partially ordered vector space (X, K). In particular, it would be nice to know the answers to following questions. Is the number of directed bands in an n-dimensional partially ordered vector space (X, K), with generating closed cone K, bounded by 2 n ? If so, is the upper bound attained if and only if (X, K) is an n-dimensional Archimedean vector lattice?