Separation properties of convexity spaces

Wiesław Kubiś Institute of Mathematics University of Silesia ul. Bankowa 14, 40-007 Katowice, Poland

                 kubis@ux2.math.us.edu.pl

October, 1997

Abstract

The purpose of this paper is to investigate some separation properties of sets with axiomatically defined convexity structures. We state a general separation theorem for pairs of convexities, improving some known results. As an application, we discuss separation properties of lattices, real vector spaces and modules.

Key words and phrases: convexity, geometrical space, Kakutani separation property, the Pasch axiom.

Mathematics Subject Classification (2000): Primary: 52A01, Secondary: 06B05.

1 Introduction

A classical theorem of Kakutani [8] says that two disjoint convex sets in a real vector space can be separated by a halfspace (i.e. a convex set with the convex complement). This theorem is also known as a geometric version of the Hahn-Banach theorem. In this paper we study the Kakutani separation property which means, roughly speaking, that disjoint sets of a certain type can be separated by disjoint complementary sets of the same type (the type will be described by means of convexity). In 1952, J.W. Ellis [4] showed an abstract version of Kakutani’s theorem, for pairs of convexities which are join-hull commutative (see the definitions below) and have a certain property, similar to the well-known Pasch axiom. The separation is realized by a set which is convex with respect to the first convexity and its complement is convex with respect to the second one. In case where both convexities are the same, Chepoi [2] showed in 1993 that the join-hull commutativity assumption is irrelevant. Our purpose is to give some sufficient conditions for the Kakutani separation property of spaces with pairs of convexities defined by finitary set operators (see the definition below). We state a general separation theorem which is improves both the results of Ellis and Chepoi. In particular, we show that Ellis’ theorem holds without the assumption of join-hull commutativity. We also characterize the Kakutani property of convexities in modules, defined by algebraic intervals [7]. This class of convexities contains the real vector space convexity, lattice convexities in Boolean algebras and other known convex structures.

2 Finitary set operators

Following van de Vel’s monograph [15], by a convexity in a set X we mean a collection G ⊂ P(X) containing ∅, X, closed under arbitrary intersections and closed under the unions of chains. The convex hull of a set A ⊂ X is the set

conv A = ⋂{G ∈ G : A ⊂ G}.

The convex hull of a set {x_{1}, …, x_{n}} is called an n-polytope and is denoted by [x_{1}, …, x_{n}]. A 2-polytope [a, b] is called the segment joining a, b. It is known [11] that conv A = ⋃{conv F : F is a finite subset of A}. A convexity G is called N-ary (N ∈ ℕ) if A ∈ G whenever conv F ⊂ A for all F ∈ [A]^{⩽N}, where [A]^{⩽N} denotes the collection of at most N-element subsets of A. A space with 2-ary convexity will be called geometrical. A convexity G in X is join-hull commutative [9] provided ⋃_{g∈G}[g, x] ∈ G whenever G ∈ G and x ∈ X. It is known [9, Th. 2] and easy to show that every join-hull commutative convexity is geometrical. For a general theory of convexity we refer to [15] and [11]. Let X be a set and let D = [X]^{⩽N} or D = [X]^{<ω}, where [X]^{<ω} denotes the collection of all finite subsets of X. Any map r: D → P(X) such that F ⊂ r(F) for all F ∈ D will be called a finitary set operator (FS-operator for short) in X. An FS-operator r: D → P(X) is N-ary provided D ⊂ [X]^{⩽N}. We say that a set G ⊂ X is r-convex if for every F ∈ D ∩ P(G) we have r(F) ⊂ G. Denote by G_{r} the collection of all r-convex subsets of X. Clearly, this is a convexity in X; G_{r} is N-ary if r is N-ary. We will say that a convexity G is defined by r if G = G_{r}. Every convexity is defined by its polytope operator I = conv | [X]^{<ω}. An N-ary convexity is defined by its N-polytope operator I_{N} = conv | [X]^{⩽N}. A set operator r is transitive provided for every y ∈ r({x}∪F) we have r({y}∪F) ⊂ r({x} ∪ F). A transitive 2-ary operator is called an interval operator (see [2]). Let r be an FS-operator. We set rA = ⋃{r(F) : F ∈ D ∩ P(A)}, r^{n+1}A = r(r^{n}A), r^{0}A = A.

Proposition 2.1. If r is a finitary set operator in X then for every A ⊂ X we have

conv A = ⋃_{n∈ℕ} r^{n}A,

where conv is the convex hull operator associated with convexity G_{r}.

Proof. Let D = dom(r), the domain of r. Denote by B the set on the right hand side. Clearly A ⊂ B and B ⊂ conv A. Let F = {a_{1}, …, a_{k}} ∈ D ∩ P(B), a_{i} ∈ r^{n_{i}}A. If we set n = max{n_{1}, …, n_{k}} then F ⊂ r^{n}A since E ⊂ rE for E ∈ D and thus r(F) ⊂ r^{n+1}A ⊂ B. Hence B is r-convex.

3 Main results

We start with a general separation theorem concerning two arbitrary convexities in a set.

Theorem 3.1. Let G and H be two convexities in a set X. The following conditions are equivalent:

(a) For every x, y, z ∈ X and finite sets S, T ⊂ X such that x ∈ conv_{G}({z} ∪ S) and y ∈ conv_{H}({z} ∪ T) it holds that conv_{G}({y} ∪ S) ∩ conv_{H}({x} ∪ T) ≠ ∅.

(b) If A ∈ G and B ∈ H are disjoint then there exist disjoint sets G ∈ G and H ∈ H such that A ⊂ G, B ⊂ H and G ∪ H = X.

Proof. (a) ⟹ (b) Let G be a maximal G-convex set containing A disjoint from B and let H be a maximal H-convex set containing B disjoint from G. We show that G ∪ H = X. Suppose otherwise, i.e. there is some z ∈ X \ (G ∪ H). By the maximality of G we get conv_{G}(G ∪ {z}) ∩ H ≠ ∅ and therefore there exists an h ∈ H and a finite set S ⊂ G with h ∈ conv_{G}({z} ∪ S). By the same argument there is a g ∈ G and a finite set T ⊂ H with g ∈ conv_{H}({z} ∪ T). By (a) we get conv_{G}({g}∪S)∩conv_{H}({h}∪T) ≠ ∅ which means that G∩H ≠ ∅; a contradiction.

(b) ⟹ (a) Let A = conv_{G}({y} ∪ S) and B = conv_{H}({x} ∪ T). Suppose that A ∩ B = ∅. By (b) there are disjoint complementary sets G ∈ G and H ∈ H with A ⊂ G and B ⊂ H. If z ∈ G then x ∈ conv_{G}({z} ∪ S) ⊂ G. This yields that z ∉ G. By the same argument z ∉ H; a contradiction.

With respect to the above theorem we introduce the following definitions.

Definition 3.2. A pair (r, s) of FS-operators in X is said to satisfy the Pasch axiom provided for each N, K < ω with [X]^{⩽N} ⊂ dom(r), [X]^{⩽K} ⊂ dom(s) and for every c, a_{1}, …, a_{N−1}, b_{1}, …, b_{K−1}, x, y ∈ X we have

x ∈ r(c, a_{1}, …, a_{N−1}) ∧ y ∈ s(c, b_{1}, …, b_{K−1}) ⟹

⟹ r(y, a_{1}, …, a_{N−1}) ∩ s(x, b_{1}, …, b_{K−1}) ≠ ∅. (P)

Here we write r(x_{1}, …, x_{N}) and s(x_{1}, …, x_{K}) instead of r({x_{1}, …, x_{N}}) and s({x_{1}, …, x_{K}}) respectively. If r = s then we say that r satisfies the Pasch axiom.

Definition 3.3. A pair of convexities (G, H) in a set X has the Kakutani separation property provided for each two disjoint sets A ∈ G and B ∈ H there exist disjoint sets G ∈ G and H ∈ H such that A ⊂ G, B ⊂ H and G ∪ H = X. If G = H then we say that the convexity space (X, G) has the Kakutani separation property.

Theorem 3.1 now says that the Kakutani separation property for two convexities is equivalent to the Pasch axiom for their polytope operators. A natural question is whether the Kakutani property holds for convexities defined by FS-operators satisfying the Pasch axiom. As we will show, the answer is affirmative for transitive set operators. We give also a similar result for arbitrary FS-operators but the Pasch axiom is then replaced with a more complicated formula.

Definition 3.4. We say that a pair (r, s) of FS-operators in X satisfies axiom (Q) provided for each natural numbers N, K with [X]^{⩽N} ⊂ dom(r), [X]^{⩽K} ⊂ dom(s) and for each b, y_{1}, …, y_{K}, h_{1}, …, h_{K}, a_{1}, …, a_{N−1} we have

b ∈ s(y_{1}, …, y_{K}) ∧ ∀i ⩽ K h_{i} ∈ r(y_{i}, a_{1}, …, a_{N−1}) ⟹

⟹ r(b, a_{1}, …, a_{N−1}) ∩ s(h_{1}, …, h_{K}) ≠ ∅. (Q)

Setting h_{i} = y_{i} for i > 1 above we see that (Q) implies the Pasch axiom.

Proposition 3.5. If (r, s) is a pair of FS-operators satisfying the Pasch axiom and such that r is transitive then (r, s) satisfy (Q).

Proof. Assume b ∈ s(y_{1}, …, y_{K}) and h_{i} ∈ r(y_{i}, a_{1}, …, a_{N−1}) for i ⩽ K. By the Pasch axiom there exists x_{1} ∈ r(b, a_{1}, …, a_{N−1})∩s(h_{1}, y_{2}, …, y_{K}) and by the transitivity of r we have r(x_{1}, a_{1}, …, a_{N−1}) ⊂ r(b, a_{1}, …, a_{N−1}). Inductively, we can find x_{2}, …, x_{N} such that x_{i} ∈ s(h_{1}, …, h_{i}, y_{i+1} …, y_{K}) and r(x_{i}, a_{1}, …, a_{N−1}) ⊂ r(b, a_{1}, …, a_{N−1}). For i = K we get r(b, a_{1}, …, a_{N−1}) ∩ s(h_{1}, …, h_{K}) ≠ ∅.

We are going to show that axiom (Q) implies the Kakutani separation property. For convenience we will use the following abbreviations: A_{r}[x] = conv_{r}(A ∪ {x}) and A_{r}^{n}[x] = r^{n}(A ∪ {x}) (the same for s). By Proposition 2.1 we have A_{r}[x] = ⋃_{n∈ℕ} A_{r}^{n}[x] (and the same for s). In the next two lemmas we assume that (r, s) is a pair of FS-operators satisfying axiom (Q).

Lemma 3.6. If H is s-convex and r(x, a_{1}, …, a_{N−1}) ∩ H ≠ ∅, then for every y ∈ H_{s}[x] we have r(y, a_{1}, …, a_{N−1}) ∩ H ≠ ∅.

Proof. The statement is clear for y ∈ H_{s}^{0}[x] = H ∪ {x}, so assume that r(y, a_{1}, …, a_{N−1})∩H ≠ ∅ whenever y ∈ H_{s}^{n}[x] and consider y ∈ H_{s}^{n+1}[x]. There are y_{1}, …, y_{K} in H_{s}^{n}[x] with y ∈ s(y_{1}, …, y_{K}), where K is such that [X]^{⩽K} ⊂ dom(s). Now, by induction hypothesis, there exist h_{i} ∈ r(y_{i}, a_{1}, …, a_{N−1}) ∩ H, i ⩽ K. Applying (Q) we have r(y, a_{1}, …, a_{N−1}) ∩ s(h_{1}, …, h_{K}) ≠ ∅. Since H is s-convex, we get r(y, a_{1}, …, a_{N−1}) ∩ H ≠ ∅.

Lemma 3.7. If G, H are such subsets of X that G is r-convex, H is s-convex and G_{r}[x] ∩ H ≠ ∅ ≠ G ∩ H_{s}[x] for some x ∈ X, then G ∩ H ≠ ∅.

Proof. We use induction. Suppose that G ∩ H ≠ ∅ whenever G is r-convex, H is s-convex and G_{r}^{n}[x] ∩ H ≠ ∅ ≠ G ∩ H_{s}[x]. Assume G_{r}^{n+1}[x] ∩ H ≠ ∅ ≠ G ∩ H_{s}[x]. This means that r(u_{1}, …, u_{N}) ∩ H ≠ ∅ for some u_{1}, …, u_{N} ∈ G_{r}^{n}[x], where N is such that [X]^{⩽N} ⊂ dom(r). Now, for i ⩽ N, we have

G_{r}^{n}[x] ∩ H_{s}[u_{i}] ≠ ∅ ≠ G ∩ (H_{s}[u_{i}])_{s}[x].

By induction hypothesis there are g_{i} ∈ G∩H_{s}[u_{i}] for i ⩽ N. Hence, as r(u_{1}, …, u_{N}) ∩H ≠ ∅ and g_{1} ∈ H_{s}[u_{1}], applying Lemma 3.6 we see that H ∩ r(g_{1}, u_{2}, …, u_{N}) ≠ ∅. Inductively, using Lemma 3.6, we infer that H ∩ r(g_{1}, …, g_{i}, u_{i+1}, …, u_{N}) ≠ ∅ for i = 1, …, N. Particularly we get H ∩ r(g_{1}, …, g_{N}) ≠ ∅ which means that G and H intersect, since G is r-convex.

Now we can state the main result.

Theorem 3.8. If (r, s) is a pair of finitary set operators satisfying axiom (Q) then the pair of convexities (G_{r}, G_{s}) has the Kakutani separation property.

Proof. Let G be a maximal r-convex set containing A disjoint from B and let H be a maximal s-convex set containing B disjoint from G. Observe that G ∪ H = X. Indeed, otherwise there exists x ∈ X \ (G ∪ H) and by the maximality of G and H we obtain G_{r}[x] ∩ H ≠ ∅ ≠ G ∩ H_{s}[x]. Hence, in view of Lemma 3.7, the sets G, H intersect; a contradiction.

4 Consequences

Theorem 4.1. Let G, H be two 2-ary convexities in a set X. The following conditions are equivalent:

(a) If a_{1} ∈ [c, a]_{G} and b_{1} ∈ [c, b]_{H} then [a, b_{1}]_{G} ∩ [b, a_{1}]_{H} ≠ ∅.

(b) If A ∈ G and B ∈ H are disjoint then there exist disjoint sets G ∈ G, H ∈ H such that A ⊂ G, B ⊂ H and G ∪ H = X.

Proof. Implication (a) ⟹ (b) follows from Theorem 3.8 and Proposition 3.5 by setting r(a, b) = [a, b]_{G} and s(a, b) = [a, b]_{H} since polytope maps are transitive. Implication (b) ⟹ (a) follows from Theorem 3.1.

J.W. Ellis has proved in [4] that the Kakutani separation property holds for two join-hull commutative convexities such that their segments satisfy condition (a) above. As every join-hull commutative convexity is 2-ary, Theorem 4.1 improves the result of Ellis. We say that two sets A, B ⊂ X are screened by C, D if A ⊂ C \ D, B ⊂ D \ C and C ∪ D = X (cf. [14]).

Theorem 4.2. Let X be a space with N-ary convexity. The following conditions are equivalent:

(i) Every two disjoint convex subsets of X can be separated by complementary halfspaces.

(ii) Every two disjoint N-polytopes can be separated by complementary halfspaces.

(iii) Every two disjoint N-polytopes can be screened by convex sets.

(iv) If x ∈ [c, a_{1}, …, a_{N−1}], y ∈ [c, b_{1}, …, b_{N−1}] then [y, a_{1}, …, a_{N−1}]∩[x, b_{1}, …, b_{N−1}] ≠ ∅.

Proof. Implication (iv) ⟹ (i) follows from Theorem 3.8 by setting r = s = I_{N} (the N-polytope map). Since implications (i) ⟹ (ii) ⟹ (iii) are trivial, it remains to show that (iii) ⟹ (iv). We use the same argument as in the proof of Theorem 3.1. Suppose (iv) fails for some x, y, c, a_{1}, …, a_{N−1} and b_{1}, …, b_{N−1}. Let C, D be two convex sets screening [y, a_{1}, …, a_{N−1}] and [x, b_{1}, …, b_{N−1}]. If c ∈ C then x ∈ [c, a_{1}, …, a_{N−1}] ⊂ C, since C is convex. Hence c ∉ C; by the same argument c ∉ D. It follows that C ∪D ≠ X; a contradiction.

V. Chepoi has stated in [2] the equivalence of (i), (ii) and the property of Npolytopes which is in fact a reformulation of axiom (Q). Thus Theorem 4.2 improves the result of Chepoi.

Theorem 4.3 (Chepoi [2]). Let X be a geometrical space defined by an interval operator I: [X]^{⩽2} → P(X). If I satisfies the Pasch axiom then X has the Kakutani separation property.

Proof. This follows immediately from Theorem 3.8 and Proposition 3.5 by setting r = s = I.

5 Examples and applications

The convexity in a lattice. Let L be a lattice; we denote by ab and a + b the infimum and the supremum of a, b in L, respectively. Set [a, b] = {x ∈ L : ab ⩽ x ⩽ a + b}. This defines a convexity in L (see [1], [13], [14]) and L is a geometrical space with such a convexity. Observe ideals and filters are convex, more generally, a convex set is an order-convex sublattice. It is easy to check that a proper halfspace is either a prime filter or a prime ideal (see [13]). Recall that a lattice L is distributive provided a(b + c) = (ab) + (ac) for every a, b, c ∈ L. Using Theorem 4.3 we are able to give a short proof of Stone-Birkhoff’s separation theorem.

Theorem 5.1. A lattice has the Kakutani property iff it is distributive.

Proof. Let L be a lattice, assume first that L has the Kakutani property. Suppose that there are a, b, c ∈ L with x = (ab) + (ac) < a(b + c) = y. Thus y ∉ [ab, ac] so there exists a halfspace H ⊂ L with y ∉ H ⊃ [ab, ac]. Observe that a ∈ L \ H since x ∈ H and y ∈ [x, a] ∩ (L \ H). Hence b, c ∈ H since ab ∈ H ∩ [a, b] and ac ∈ H ∩ [a, c]. Now b + c ∈ H and y ∈ [x, b + c] ⊂ H which gives a contradiction. Now assume that L is distributive and consider a, a_{1}, b, b_{1}, c ∈ L with a_{1} ∈ [c, a] and b_{1} ∈ [c, b]. We have ac ⩽ a_{1} ⩽ a + c and bc ⩽ b_{1} ⩽ b + c, whence

a_{1}b ⩽ (a + c)b = (ab) + (cb) ⩽ a + b_{1}. (1)

Similarly ab_{1} ⩽ a_{1} + b. We set x = (a + b_{1})(a_{1} + b). Clearly x ⩽ a_{1} + b and by (1) we get x ⩾ a_{1}b(a_{1} + b) = a_{1}b. Hence x ∈ [a_{1}, b]. By the same argument x ∈ [a, b_{1}]. Now Theorem 4.3 implies that L has the Kakutani property.

Geometrical modules. Let us consider a ring R (with unity, but not necessarily commutative). Following Jamison [7] we say that a subset J of R is an algebraic interval provided

(i) 0, 1 ∈ J and

(ii) α, β, γ ∈ J implies γα + (1 − γ)β ∈ J.

Fix an R-module M. For a, b ∈ M we set

[a, b]_{J} = {λa + (1 − λ)b : λ ∈ J}.

This defines a 2-ary convexity G_{J} in M; elements of G_{J} will be called J-convex. Observe that (i) and (ii) imply that the set [a, b]_{J} defined above is in fact the segment joining a, b with respect to G_{J}. Any module over a ring R with such defined convexity will be called a geometrical module over (R, J). In particular R alone is a geometrical module over itself. Observe that J is the segment joining 0, 1 ∈ R. We shall state an algebraic condition for geometrical modules equivalent to the Kakutani property. First we present some examples of geometrical modules.

Examples 5.2. (a) Let ℝ denote the field of reals. If J = [0, 1] then the convexity G_{J} in a real vector space is the usual one. If J = [0, 1] ∩ ℚ then it is the rational convexity. Finally, J = \{\frac{k}{2^{n}} ∈ [0, 1] : k, n ∈ ℕ\} defines the Jensen convexity (see also [3], [5]).

(b) Let B be a Boolean algebra and let J = B. Denote by △ the symmetric difference in B. Then (B, △, ∗, 0, 1) is a (Boolean) ring. One can easily check that [a, b]_{J} = {x ∈ B : ab ⩽ x ⩽ a + b}. Hence this convexity is the same as the convexity defined by the lattice structure in B.

(c) Let R be the ring of all real measurable functions defined on a measurable space (T, 𝔐). We set J = {χ_{A} : A ∈ 𝔐} where χ_{A} denotes the characteristic function of A. It is easily seen that χ_{C}χ_{A}+(1−χ_{C})χ_{B} = χ_{G} where G = (A∩C)∪(B\C), hence J is an algebraic interval. A J-convex set is called decomposable; decomposable subsets of L^{1}(μ), the space of μ-integrable real functions, were introduced in [6].

Theorem 5.3. Let R be a ring and J ⊂ R. The following conditions are equivalent:

(a) Every geometrical module over (R, J) has the Kakutani property.

(b) J (with the relative convexity) has the Kakutani property.

(c) For each α, β ∈ J there exists γ ∈ J with (1 − α)β = γ(1 − αβ).

Proof. (a) ⟹ (b) As we have already mentioned, J is a convex subset of R. Hence disjoint convex subsets of J are convex in R and the Kakutani property of J follows from that of R.

(b) ⟹ (c) Fix α, β ∈ J. Notice that αβ ∈ [0, β]_{J} ⊂ J. Suppose that β ∉ [1, αβ]_{J}. As {β} and [1, αβ]_{J} are convex and disjoint, there exists a halfspace H ⊂ J with β ∉ H and [1, αβ]_{J} ⊂ H. If 0 ∈ H then β = β · 1 + (1 − β) · 0 ∈ H, if 0 ∉ H then αβ = αβ + (1 − α) · 0 ∉ H. Both cases yield a contradiction. It follows that β ∈ [1, αβ]_{J}, i.e. there exists γ ∈ J with β = γ + (1 − γ)αβ. Modifying this expression we get (1 − α)β = γ(1 − αβ).

(c) ⟹ (a) In view of Theorem 4.3 we must verify the Pasch axiom. Fix a geometrical module M over (R, J) and a, a_{1}, b, b_{1}, c ∈ M such that a_{1} ∈ [c, a]_{J} and b_{1} ∈ [c, b]_{J}. We have

a_{1} = αa + (1 − α)c, b_{1} = βb + (1 − β)c,

for some α, β ∈ J. Let γ ∈ J be as in condition (c), i.e.

β = αβ + γ(1 − αβ) = (1 − γ)αβ + γ. (1)

We set

δ = (1 − γ)α. (2)

Observe that δ ∈ [0, α]_{J} ⊂ J and γ = (1 − δ)β. Now take

x_{1} = (1 − γ)a_{1} + γb, x_{2} = (1 − δ)b_{1} + δa.

Notice that x_{1} ∈ [a_{1}, b]_{J} and x_{2} ∈ [a, b_{1}]_{J}. It remains to check that x_{1} = x_{2}. Let us compute the difference x_{1} − x_{2}. We have

x_{1} − x_{2} = (1 − γ)(αa + (1 − α)c) + γb − (1 − δ)(βb + (1 − β)c) − δa
= δa + (1 − γ)(1 − α)c + γb − γb − (1 − δ)(1 − β)c − δa
= ((1 − γ)(1 − α) − (1 − δ)(1 − β))c.

Now, applying (1) and (2) we see that

(1 − δ)(1 − β) = 1 − δ − β + (1 − γ)αβ = 1 − δ − β + β − γ
= 1 − γ − (1 − γ)α = (1 − γ)(1 − α).

Hence x_{1} = x_{2} and this completes the proof.

Corollary 5.4. (a) If J = F ∩ [0, 1] where F is a subfield of reals then every real vector space with J-convexity has the Kakutani property.

(b) (Palés [10]) The Jensen convexity in ℝ does not have the Kakutani property.

(c) For every measure μ, the ring of μ-measurable real functions, with the convexity of decomposable sets, has the Kakutani property. The same is true for L^{1}(μ).

Proof. (a) is clear, by condition (c) of Theorem 5.3.

(b) follows from the equality (1 − \frac{1}{2})\frac{1}{2}/(1 − \frac{1}{2} \frac{1}{2}) = \frac{1}{3}.

(c) Equality (1 − χ_{A})χ_{B} = γ(1 − χ_{A}χ_{B}) is valid for γ = χ_{B}. Moreover, L^{1}(μ) is a decomposable subset of the ring of all μ-measurable functions.

References

[1] G. Birkhoff, S.A. Kiss, A ternary operation in distributive lattices, Bull. Amer. Math. Soc. 53 (1947) 749-752.

[2] V. Chepoi, Separation of two convex sets in convexity structures, J. Geom. 50 (1994) 30-51.

[3] H.G. Eggleston, Convexity, Cambridge University Press, Cambridge 1969.

[4] J.E. Ellis, A general set-separation theorem, Duke Math. J. 19 (1952) 417-421.

[5] J.W. Green, W. Gustin, Quasiconvex sets, Canad. J. Math. 2 (1950) 489-507.

[6] F. Hiai, H. Umegaki, Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate Anal. 7 (1977) 149-182.

[7] R.E. Jamison, Functional representation of algebraic intervals, Pacific J. Math. Vol. 53, No. 2 (1974) 399-423.

[8] S. Kakutani, Ein Beweis des Sätzes von Edelheit über konvexe Mengen, Proc. Imp. Acad. Tokyo 13 (1937) 93-94.

[9] D.C. Kay, E.W. Womble, Axiomatic convexity theory and relationships between the Carathéodory, Helly, and Radon numbers, Pacific J. Math. 38/2 (1971) 471-485.

[10] Zs. Palés, On the separation of midpoint convex sets, C. R. Math. Rep. Acad. Sci. Canada Vol. VIII, No. 5 (1986) 309-312.

[11] V.P. Soltan, An Introduction to the Axiomatic Theory of Convexity, (Russian) Kishinev 1984.

[12] M.H. Stone, Topological representations of distributive lattices and Brouwerian logics, Čas. pro Pěst. Mat. a Fys. 67 (1937) 1-25.

[13] J.C. Varlet, Remarks on distributive lattices, Bull. Acad. Polon. Sci. Vol. 23, No. 11 (1975) 1143-1147.

[14] M. van de Vel, Binary convexities and distributive lattices, Proc. London Math. Soc. (3) 48 (1984) 1-33.

[15] M. van de Vel, Theory of convex structures, North-Holland, Amsterdam