Characterizations of Convex spaces and Anti-matroids via Derived Operators

Abstract In this paper we use the notion of derived sets to study convex spaces. By axiomatizing the derived sets on convex spaces, we define c-derived operators and restricted c-derived operators. Results show that convex structures can be characterized in terms of c-derived operators. Furthermore, the link between c-derived operators and Shi’s m-derived operators is studied. Specifically, it is proved that a c-derived operator is an m-derived operator if and only if it satisfies the Exchange Law. At last, we show an application of c-derived operators to anti-matroids.


Introduction
Convexity theory has been accepted to be increasingly important in recent years in the study of extremum problems of applied mathematics. In fact, convex structure exists in many mathematical research areas, such as lattices [1,2], algebras [3,4], metric spaces [5], graphs [6][7][8] and topological spaces [9,10]. In 1993, M. van de Vel collected the convexity theory systematically in the famous book [11].
A convex structure [11] on a set is a family of subsets which contains the empty set and is closed under arbitrary intersections and directed unions. Often it is more convenient not to describe the family of convex sets directly, and thus some other characterizations of convex structures become especially important, which can be found in [11] consisting of hull operators, restricted hull operators, betweenness relations and independence structures.
The notion of derived sets is rst introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line. In mathematics, more specially in point-set topology, the derived sets of a subset S of a topological space is the set of all limit points of S. Various topological notions can be characterized in terms of derived sets (refer to [12,13]): (T1) A subset is closed precisely when it contains its derived set. (T2) Two subsets are separated if and only if they are disjoint and each is disjoint from the other's derived set. (T3) A bijection between two topological spaces is a homeomorphism if and only if the derived set of the image of any subset is the image of the derived set of that subset.
Conversely, a mapping co : P(X) −→ P(X) with the properties (AC1)-(AC4), determines a convex structure C on X de ned as follows: C = A ∈ P(X) | co(A) = A .

Further, the relationship between hull operators and convex structures on a given set is bijective.
De nition 2.4 ([15]). A convex structure C on a nite set X is called (1) a matroid provided its hull operator co satis es the Exchange Law: if A ⊆ X and p, q ∈ X − co(A) with p ≠ q, then p ∈ co({q} ∪ A) implies q ∈ co({p} ∪ A); (2) a anti-matroid (or a convex geometry) provided its hull operator co satis es the Anti-Exchange Law: if Proposition 2.5 ([16]). Let C be an anti-matroid on a nite set X and let A, De nition 2.6 ( [16]). Let C be a convex set in a convex space (X, C ). A subset A ⊆ C is said to generate C if co(A) = C. More specially, if A is the minimal generating set, it is called the generator of C. When there is only a single generating set for any convex set in C , we say that the convex structure C is uniquely generated.
Proposition 2.7 ([16]). Let X be a nite nonempty set. Then a convex structure C on X is anti-matroid if and only if it is uniquely generated.
De nition 2.8 ([11]). Let f : (X, C X ) −→ (Y , C Y ) be a mapping between convex spaces. Then f is called Theorem 2.9 ([11]). Let f : (X, C X ) −→ (Y , C Y ) be a mapping between two convex spaces. Then the following conditions are equivalent.
The category of convex spaces and CP mappings is denoted Conx.

Derived operators
In this section, the notion of c-derived operators is presented. Further, it is proved that every c-derived operator can induce a convex structure.
De nition 3.1. Let X be a set. A mapping d : P(X) −→ P(X) is called a convexly derived operator (c-derived operator for short) on X provided d satis es the following conditions: We call the pair (X, d) a convexly derived space (c-derived space for short) if d is a c-derived operator on X.

De nition 3.3. A mapping
Proof. The necessity is obvious. For su ciency, take any A ⊆ X. By (CD4) and (CD2 * ), we obtain The proof is completed.
The following conclusion is straightforward.
be two DP mappings between cderived spaces. Then the composite mapping g • f is also a DP mapping.
The category of c-derived spaces and DP mappings is denoted by Deri. Proposition 3.6. Let d be a c-derived operator on X. Then the family C d ⊆ P(X) de ned by Proof. The veri cations of (CS1) and (CS2) are straightforward. For (CS3), take any directed family

Proposition 3.7. Let d be a c-derived operator on X, let C d ⊆ P(X) be the convex structure induced by d and let co d be the hull operator on
Proof. It's trivial by the Proposition 3.7.
By Proposition 3.6 and Proposition 3.8, we obtain a functor F :Conx−→Deri de ned by

Derived Operators Induced by Convexities
In this part, we show that every c-derived operator can be induced by a convex structure. Moreover, it is proved that the category of c-derived spaces is isomorphic to that of convex spaces.
De nition 4.1. Let (X, C ) be a convex space. Then for each A ⊆ X, the set d C (A) de ned as follows: is called the c-derived set of A.

Proposition 4.2.
Let (X, C ) be a convex space with the corresponding hull operator co. Then the following statements hold for any A ∈ P(X).
It follows x ∈ co(F − {x}) for some F ∈ P n (A). This means x ∈ K. We obtain d C (A) ⊆ K. Therefore Moreover, an easy induction can obtain that (2) It su ces to prove co( Take any x ∈ co(A) and x ∉ A. Then by (AC4) of Proposition 2.3, we obtain x ∈ co(F) for some F ∈ P n (A). Note that x ∉ A, which means x ∈ co(F) ∩ (X − F). By the conclusion of (1), we obtain x ∈ d C (A).
Next, we will verify that the operator d C is a c-derived operator on X. Before proving this, the following lemma is necessary.
. Then there exists a nite subset F ⊆ d C (A) satisfying x ∈ co(F). By (1), there exists G ∈ P n (A) satisfying co(F) ⊆ co(G), which means x ∈ co(G − {x}). This shows that x ∈ d C (A), a contradiction.
(CD4) On one hand, the inequality holds obviously. On the other hand, take any Proof. It su ces to prove f (d X (A)) ⊆ f (A) ∪ d Y (f (A)) for all A ⊆ X. Since f : (X, C X ) −→ (Y , C Y ) is CP and Theorem 2.9, we have f (co X (A)) ⊆ co Y (f (A)). It follows that The proof is completed.  Proof. We need to prove G • F = I Deri and F • G = I Conv . It su ces to verify (1) d C d = d and (2) C d C = C for any c-derived space (X, d) and convex space (X, C ).

For (1), take any
For (2), let co be the hull operator on (X, C ). By Lemma 4.6, we have If y ≠ x and y ∈ A, then y ∈ A − {x} ⊆ co(A − {x}). Now assume y ≠ x, y ∈ d C (A) and y ∉ A. It follows from y ∈ d C (A) that there exists F ∈ P n (A) such that y ∈ co(F).   (1) Let P be a poset. A subset C is called order convex provided z ∈ C whenever x ≤ z ≤ y and x, y ∈ C.
Then the family C (P) of all order convex sets forms a convex structure on P, and it is trivial to check that for every subset A of P, the c-derived set (2) Let X = {a i | i = , , , , } be a metric space, and the metric δ on X is de ned as gure 1. Note that  δ(a , a ) = δ(a , a ) δ(a, x) + δ(x, b) = δ(a, b) and a, b ∈ C.  (3) Let V be a vector space over a eld K and let V be the set V minus the zero vector . A nonempty set C ⊆ V is called linear convex provided s · p + t · q ∈ C whenever p, q ∈ C and s, t ∈ K.
Then the family of all linear convex sets forms a convex structure on V , and it is trivial to check that for any subset A of V , co(A) = n i= t i · p i | p , p , · · · pn ∈ A, t , t , · · · tn ∈ K , and hence x ∈ d(A) if and only if x = n i= t i · p i for some p , p , · · · pn ∈ A − {p}, t , t , · · · tn ∈ K.

Restricted c-derived operators
In convexity theory, a notable result is that every convex structure can be completely determined by the polytopes (the hull of nite sets). This property entails a new operator, called restricted hull operator, by restricting the hull operator to the family of all nite sets. Motivated by this, we present the notion of restricted c-derived operators, and establish its relationship to convex structures.
A restricted hull operator [11] is a mapping h : P n (X) −→ P(X) satisfying the following conditions: (H1) h(∅) = ∅; (H2) for any F ∈ P n (X), F ⊆ h(F); (H3) for any F, G ∈ P n (X), It is known that every restricted hull operator can uniquely determine a convex structure [11].
De nition 5.1. A mapping d : P n (X) −→ P(X) is called a restricted c-derived operator provided for any F, G ∈ P n (X), it satis es the following conditions: Proof. The veri cations of (H1) and (H2) are trivial. For (H3), Assume that F, G ∈ P n (X) and Lemma 5.3. Every restricted hull operator h is order-preserving, that is, for any F, G ∈ P n (X), Proposition 5.4. Let h be a restricted hull operator on X. Then the mapping d h : P n (X) −→ P(X) de ned by is a restricted c-derived operator.

Theorem 5.5. The relationship between restricted c-derived operators and restricted hull operators is bijective.
Proof. Let (X, d) be a c-derived space and let (X, h) be a restricted hull space. It su ces to prove (1) (2) Take any F ∈ P n (X). On one hand, we have On the other hand, take any x ∈ h(F). If x ∈ F, then we are done. So assume x ∉ F, and Theorem 5.6. Let d be a restricted c-derived operator on X. Then there is precisely one convex structure on X with a c-derived operator equal to d on P n (X). Conversely, the c-derived operator of any convex structure on X satis es the conditions (RD1)-(RD4).
Next we will show that the m-derived operators are exactly the c-derived operators satisfying the Exchange Law.

Theorem 6.2. A c-derived operator d is an m-derived operator on a nite set E if and only if it satis es the following Exchange Law:
Proof. Necessity. It su ces to verify (CD5). Since q ∉ C (otherwise p ∈ d(C)),

Proof. Necessity. Note that co(
Su ciency. Take any p ∈ d (C ∪ {q}) − (C ∪ d(C)). If q ∈ d(C), then we are done. So assume q ∉ d(C). Proposition 6.6. Let (E, C ) be anti-matroid. Then for any A ⊆ E and C ∈ C , the following statements hold.
The following result shows that a convex set is completely determined by the complement of its c-derived set, and the proof is trivial by Proposition 2.7 and Proposition 6.6. Theorem 6.7. If C is a convex structure on a nite set E, then the following statements are equivalent.