Characterizations of Convex spaces and Anti-matroids via Derived Operators

Fanhong Chen; Chong Shen

doi:10.1515/math-2019-0022

Article Open Access

Characterizations of Convex spaces and Anti-matroids via Derived Operators

Fanhong Chen and Chong Shen

Published/Copyright: May 11, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

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.

Keywords: convex space; derived operator; m-derived operator; matroid; anti-matroid

MSC 2010: 52A01; 14P25

1 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 first 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]):

A subset is closed precisely when it contains its derived set.
Two subsets are separated if and only if they are disjoint and each is disjoint from the other’s derived set.
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.
Any Polish space can be written as the union of a countable set and a perfect set, where a set is called perfect if its derived set coincides with itself.

As a set-structure that is similar to topologies, a natural idea is whether convex structures can be characterized by derived operators. Another motivation comes from m-derived operators [14] introduced by Xin and Shi in 2010 which provide a new description for a matroid. As we know, a convex structure is a matroid if and only if its hull operator satisfies the Exchange Law. This encourages us to bring the notion of derived operators from matroids to convex structures directly, which can be regarded as an extension of m-derived operators. In this paper, we introduce the notion of derived operators defined in convex spaces, called c-derived operators, which turns out to be important in the study of both convex spaces and anti-matroids.

The layout of the paper is organized as follows. In Section 2, some preliminaries on convex structures and (anti-) matroids are introduced. In Sections 3 and 4, we introduce and study the notion of c-derived operators, and construct the isomorphism between c-derived operators and convex structures. Also, we proved that CP mappings can be completely described in terms of c-derived operators. In Section 5, the notion of restricted c-derived operators is introduced, which is isomorphic to convex structures. At last, the link between c-derived operators and m-derived operators are studied, and also some equivalent descriptions of anti-matroids is provided.

2 Preliminary

In this section, we shall review some basic concepts and results on the convex structures and (anti-) matroids. For undefined notions in this paper, the reader can refer to [11, 15].

Let X be a nonempty set, let 𝓟(X) denote the power set of X, and 𝓟_fin(X) the family of all finite subsets of X. A family {D_i | i ∈ I} ⊆ 𝓟(X) is called directed if for each pairs i₁, i₂ ∈ I, there exists an i₃ ∈ I such that D_i₁ ⊆ D_i₃ and D_i₂ ⊆ D_i₃. It is trivial that 𝓟_fin(X) is a directed family.

Definition 2.1

([11]). A subset 𝒞 of 𝓟(X) is called a convex structure, if it satisfies the following conditions:

∅, X ∈ 𝒞;
if {A_i | i ∈ I} ⊆ 𝒞, then ⋂_{i ∈ I} A_i ∈ 𝒞;
if {D_i | i ∈ I} ⊆ 𝒞 is directed, then ⋃_{i ∈ I} D_i ∈ 𝒞.

The pair (X, 𝒞) is called a convex space if 𝒞 is a convex structure on X, and each A ∈ 𝒞 a convex set.

Definition 2.2

([11]). Let (X, 𝒞) be a convex space. For each A ∈ 𝓟(X), define

co(A)=⋂{B ∈ C∣A⊆B}.

Then co(A) is the least element of 𝒞 that contains A, called the (convex) hull of A. The operator co is called the hull operator on (X, 𝒞).

Convex structures can be characterized by hull operators.

Proposition 2.3

([11]). Let co be the hull operator on (X, 𝒞). Then it satisfies the following properties:

Normalization Law: co(∅) = ∅;
Extensive Law: A ⊆ co(A);
Idempotent Law: co(co(A)) = co(A);
Algebraic Law: co(A) = ⋃{co(F) | F ∈ 𝓟_fin(A)}.

Conversely, a mapping co : 𝓟(X) ⟶ 𝓟(X) with the properties (AC1)–(AC4), determines a convex structure 𝒞 on X defined as follows:

C=A ∈ P(X)∣co(A)=A.

Further, the relationship between hull operators and convex structures on a given set is bijective.

Definition 2.4

([15]). A convex structure 𝒞 on a finite set X is called

a matroid provided its hull operator co satisfies 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);
a anti-matroid (or a convex geometry) provided its hull operator co satisfies the Anti-Exchange Law: if A ⊆ X and p, q ∈ X – co(A) with p ≠ q, then p ∈ co({q} ∪ A) implies q ∉ co({p} ∪ A).

Proposition 2.5

([16]). Let 𝒞 be an anti-matroid on a finite set X and let A, B ⊆ X. If co(A) = co(B) = C, then co(A ∩ B) = C.

Definition 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 finite nonempty set. Then a convex structure 𝒞 on X is anti-matroid if and only if it is uniquely generated.

Definition 2.8

([11]). Let f : (X, 𝒞_X) ⟶ (Y, 𝒞_Y) be a mapping between convex spaces. Then f is called

convex structure-preserving (CP, in short) provided B ∈ 𝒞_Y implies f⁻¹(B) ∈ 𝒞_X;
convex-to-convex (CC, in short) provided A ∈ 𝒞_X implies f(A) ∈ 𝒞_Y.

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.

f is CP (resp., CC).
For any F ∈ 𝓟_fin(X), f(co_X(F)) ⊆ co_Y(f(F)) (resp., co_Y(f(F)) ⊆ f(co_X(F))).
For any A ∈ 𝓟(X), f(co_X(A)) ⊆ co_Y(f(A)) (resp., co_Y(f(A)) ⊆ f(co_X(A))).

The category of convex spaces and CP mappings is denoted Conx.

3 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.

Definition 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 satisfies the following conditions:

Normalization Law: d(∅) = ∅;
Representative Law: x ∈ d(A) implies x ∈ d(A − {x});
Idempotent Law: d(d(A) ∪ A) ⊆ d(A) ∪ A;
Algebraic Law: d(A) = ⋃{d(F) ∣ F ∈ 𝓟_fin(A)}.

We call the pair (X, d) a convexly derived space (c-derived space for short) if d is a c-derived operator on X.

Proposition 3.2

Conditions (CD2) and (CD4) can be replaced by

A ⊆ B implies d(A) ⊆ d(B);
d(A) = ⋃{d(F)∩ (X − F) ∣ F ∈ 𝓟_fin(A)}.

Proof

Necessity. By (CD4), the verification of (CD2^∗) is straightforward. It suffices to verify (CD4^∗). Take any x ∈ d(A). By (CD2) and (CD4), we obtain

x∈d(A−{x})=⋃{d(F)∣F∈Pfin(A−{x})}.

Then there exists G ∈ 𝓟_fin(A − {x}) such that x ∈ d(G). This implies

x∈d(G)∩(X−G)⊆⋃{d(F)∩(X−F)∣F∈Pfin(A)}.

Therefore, d(A) ⊆ ⋃{d(F)∩ (X − F) ∣ F ∈ 𝓟_fin(A)}. The opposite inclusion holds obviously. Thus (CD4^∗) holds.

Sufficiency. (CD2) Take any x ∈ d(A). By (CD4^∗), there exists F ∈ 𝓟_fin(A) such that x ∈ d(F)∩ (X − F). This means x ∈ d(F − {x}). By (CD2^∗), we have x ∈ d(A − {x}). Thus (CD2) holds.

(CD4) By using (CD4^∗), we obtain

d(A)=⋃{d(F)∩(X−F)∣F∈Pfin(A)}⊆⋃{d(F)∣F∈Pfin(A)}.

It follows from (CD2^∗) that ⋃_{F ∈ 𝓟_fin(A)}d(F) ⊆ d(A). Thus d(A) = ⋃{d(F) ∣ F ∈ 𝓟_fin(A)}. Therefore (CD4) holds.□

Definition 3.3

A mapping f : (X, d_X) ⟶ (Y, d_Y) between c-derived spaces is called c-derived-preserving (DP for short) provided that

∀A⊆X,f(dX(A))⊆f(A)∪dY(f(A)).

Proposition 3.4

Let f : (X, d_X) ⟶ (Y, d_Y) be a mapping between c-derived spaces. Then f is a DP mapping if and only if f(d_X(F)) ⊆ f(F) ∪ d_Y(f(F)) for all F ∈ 𝓟_fin (X).

Proof

The necessity is obvious. For sufficiency, take any A ⊆ X. By (CD4) and (CD2^∗), we obtain

f(dX(A))=f⋃{dX(F)∣F∈Pfin(A)}=⋃{f(dX(F))∣F∈Pfin(A)}⊆⋃{f(F)∪dY(f(F))∣F∈Pfin(A)}⊆f(A)∪dY(f(A)).

The proof is completed.□

The following conclusion is straightforward.

Proposition 3.5

Let f : (X, d_X) ⟶ (Y, d_Y) and g : (Y, d_Y) ⟶ (Z, d_Z) be two DP mappings between c-derived 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) defined by

Cd=C⊆X∣d(C)⊆C

forms a convex structure on X. Moreover, 𝒞_d = {d(A) ∪ A ∣ A ⊆ X}.

Proof

The verifications of (CS1) and (CS2) are straightforward. For (CS3), take any directed family {D_i ∣ i ∈ I} ⊆ 𝒞_d. Then d(D_i) ⊆ D_i for all i ∈ I. Furthermore, by (CD4), we have

d⋃i∈IDi=⋃{d(F)∣F⊆⋃i∈IDi}=⋃{d(F)∣∃i∈I,F⊆Di}=⋃i∈I⋃{d(F)∣F⊆Di}=⋃i∈Id(Di)⊆⋃i∈IDi.

This shows that ⋃_{i ∈ I}D_i ∈ 𝒞_d.

For convenience, denote Cd∗ := {A ∪ d(A) ∣ A ⊆ X}. It remains to show 𝒞_d = Cd∗ . First by (CD3), we have d(A ∪ d(A)) ⊆ A ∪ d(A) for all A ∈ 𝓟(X), which implies that Cd∗ ⊆ 𝒞_d. In addition, if A ∈ 𝒞_d, then d(A) ⊆ A, implying that A = A ∪ d(A) ∈ Cd∗ . It follows that 𝒞_d ⊆ Cd∗ .□

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 𝒞_d. Then co_d(A) = d(A) ∪ A for all A ∈ 𝓟(X).

Proof

On one hand, since A ⊆ d(A) ∪ A ∈ 𝒞_d, it follows that co_d(A) = ⋂{C ∈ 𝒞_d ∣ A ⊆ C} ⊆ d(A) ∪ A. On the other hand, if C ∈ 𝒞_d satisfying A ⊆ C, then d(C) ⊆ C, implying that d(A) ∪ A ⊆ d(C) ∪ C = C. Hence co_d(A) = d(A) ∪ A.

Proposition 3.8

Let f : (X, d_X) ⟶ (Y, d_Y) be a DP mapping between c-derived spaces. Then f : (X, 𝒞_d_X) ⟶ (Y, 𝒞_d_Y) is a CP mapping.

Proof

It’s trivial by the Proposition 3.7.□

By Proposition 3.6 and Proposition 3.8, we obtain a functor 𝔽 : Conx ⟶ Deri defined by

F(X,d)=(X,Cd)andF(f)=f.

4 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.

Definition 4.1

Let (X, 𝒞) be a convex space. Then for each A ⊆ X, the set d_𝒞(A) defined as follows:

dC(A)=x∈X∣x∈co(A−{x})

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).

dC(A)=x∈X∣∀C∈C,A−{x}⊆C⇒x∈C=x∈X∣∃F∈Pfin(A),x∈co(F−{x})={x∈X∣∃F∈Pfin(A),x∈co(F)∩(X−F)}.
co(A) = d_𝒞(A) ∪ A.

Proof

First, it is trivial that d_𝒞(A) = {x ∈ X ∣ ∀ C ∈ 𝒞, A − {x} ⊆ C ⇒ x ∈ C}. Let

K=x∈X∣∃F∈Pfin(A),x∈co(F−{x}).

Now we prove d_𝒞(A) = K. Take any x ∈ K. Then there exists F ∈ 𝓟_fin(A) such that x ∈ co(F − {x}). For each C ∈ 𝒞, if A − {x} ⊆ C, then F − {x} ⊆ C. It follows x ∈ co(F − {x}) ⊆ C. This means x ∈ d_𝒞(A) and thus K ⊆ d_𝒞(A). Conversely, take any x ∈ d_𝒞(A). Note that

A−{x}⊆⋃{co(F−{x})∣F∈Pfin(A)}∈C.

It follows from co(A − {x}) ⊆ ⋃{co(F − {x}) ∣ F ∈ 𝓟_fin(A)}. Take any x ∈ d_𝒞(A). Then

x∈⋃{co(F−{x})∣F∈Pfin(A)}.

It follows x ∈ co(F − {x}) for some F ∈ 𝓟_fin(A). This means x ∈ K. We obtain d_𝒞(A) ⊆ K. Therefore

dC(A)=x∈X∣∃F∈Pfin(A),x∈co(F−{x}).

Moreover, an easy induction can obtain that

K={x∈X∣∃F∈Pfin(A),x∈co(F)∩(X−F)}.
It suffices to prove co(A) ⊆ d_𝒞(A) ∪ A. Take any x ∈ co(A) and x ∉ A. Then by (AC4) of Proposition 2.3, we obtain x ∈ co(F) for some F ∈ 𝓟_fin(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.

Lemma 4.3

Let (X, 𝒞) be a convex space and let d_𝒞 be the operator induced by Definition 4.1. Then the following statements hold for any A ∈ 𝓟(X).

If F ⊆ d_𝒞(A) is finite, then F ⊆ co(G) for some G ∈ 𝓟_fin(A).
d_𝒞(d_𝒞(A)) ⊆ d_𝒞(A) ∪ A.

Proof

For each x ∈ F, we have x ∈ d_𝒞(A) which means x ∈ co(F_x) for some F_x ∈ 𝓟_fin(A). Let G = ⋃_x∈F F_x. Obviously, G is a finite subset of A satisfying F ⊆ co(G).
Assume x ∉ d_𝒞(A) ∪ A and x ∈ d_𝒞(d_𝒞(A)). Then there exists a finite subset F ⊆ d_𝒞(A) satisfying x ∈ co(F). By (1), there exists G ∈ 𝓟_fin(A) satisfying co(F) ⊆ co(G), which means x ∈ co(G − {x}). This shows that x ∈ d_𝒞(A), a contradiction.□

Proposition 4.4

For a convex space (X, 𝒞), the operator d_𝒞 is a c-derived operator on X.

Proof

The verification of (CD1) is trivial. It suffices to verify (CD2)–(CD4).

(CD2) Take any x ∈ d_𝒞(A). By Proposition 4.2, we have x ∈ co(F − {x}) for some F ∈ 𝓟_fin(A). Let E = F − {x}. It follows that E ∈ 𝓟_fin(A − {x}) and x ∈ co(E) = co(E − {x}). Hence x ∈ d_𝒞(A − {x}).

(CD3) Take any x ∈ d_𝒞(d_𝒞(A) ∪ A) and x ∉ A. Then by Proposition 4.2, there exists a finite F ⊆ d_𝒞(A) ∪ A such that x ∈ co(F)∩ (X − F). If F ⊆ A, then x ∈ d_𝒞(A). The proof is completed. Otherwise, let F₁ = F ∩ d_𝒞(A) and let F₂ = F ∩ A. Then F₁ is a nonempty finite set and F = F₁ ∪ F₂. Since F₁ ⊆ d_𝒞(A) and Lemma 4.3, there exists G ∈ 𝓟_fin(A) satisfying F₁ ⊆ co(G). It follows that x ∈ co(F) ⊆ co(co(G) ∪ F₂) = co(G ∪ F₂). Note that G ∪ F₂ ∈ 𝓟_fin(A) and x ∉ G ∪ F₂, which means x ∈ d_𝒞(A).

(CD4) On one hand, the inequality

⋃{dC(F)∣F∈Pfin(A)}⊆dC(A)

holds obviously. On the other hand, take any x ∈ d_𝒞(A). Then there exists E ∈ 𝓟_fin(A) satisfying x ∈ co(E − {x}). This implies x ∈ d_𝒞(E) ⊆ ⋃{d_𝒞(F) ∣ F ∈ 𝓟_fin(A)}.□

Proposition 4.5

Let f : (X, 𝒞_X) ⟶ (Y, 𝒞_Y) be a CP mapping between c-derived spaces. Then f : (X, d_{𝒞_X}) ⟶ (Y, d_{𝒞_Y}) is a DP mapping.

Proof

It suffices 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

f(dX(A))⊆f(dX(A)∪A)=f(coX(A))⊆coY(f(A))=f(A)∪dY(f(A)).

The proof is completed.□

By Proposition 4.4 and Proposition 4.5, we obtain a functor 𝔾 : Conv ⟶ Deri defined by

G(X,C)=(X,dC)andG(f)=f.

Lemma 4.6

Let (X, 𝒞) be a convex space and (X, d_𝒞) be the c-derived space generated by 𝒞. Then we have

dC(A)⊆Aifandonlyifco(A)⊆A.

Proof

Assume co(A) ⊆ A. Take any x ∈ d_𝒞(A). Then there exists F ∈ 𝓟_fin(A) satisfying x ∈ co(F) ⊆ co(A) ⊆ A. Conversely, Assume d_𝒞(A) ⊆ A. Then for each x ∈ co(A)∩ (X- A), there exists F ∈ 𝓟_fin(A) satisfying x ∈ co(F)∩ (X − F). This shows that x ∈ d_𝒞(A) ⊆ A, a contradiction.□

Now we give the main result in this section.

Theorem 4.7

The category Deri is isomorphic to Conv.

Proof

We need to prove 𝔾 ∘ 𝔽 = 𝕀_Deri and 𝔽 ∘ 𝔾 = 𝕀_Conv. It suffices 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 A ⊆ X. Since co_d(A) = d(A) ∪ A, we have

dCd(A)={x∈X∣∃F∈Pfin(A),x∈cod(F−{x})}={x∈X∣∃F∈Pfin(A),x∈d(F−{x})∪(F−{x})}={x∈X∣∃F∈Pfin(A),x∈d(F−{x})}={x∈X∣∃F∈Pfin(A),x∈d(F)}=⋃{d(F)∣F∈Pfin(A)}=d(A)}.

For (2), let co be the hull operator on (X, 𝒞). By Lemma 4.6, we have

CdC={C⊆X∣dC(C)⊆C}={C⊆X∣co(C)⊆C}=C.

□

Proposition 4.8

Let (X, 𝒞) be a convex space and let A ∈ 𝓟(X). Then x ∈ d_𝒞(A) if and only if x ∈ co(A) and co(A) = co(A − {x}).

Proof

Necessity. Assume that x ∈ d_𝒞(A). Clearly, x ∈ co(A). It remains to show A ∪ d_𝒞(A) ⊆ co(A − {x}). Take any y ∈ A ∪ d_𝒞(A). If y = x, then y ∈ d_𝒞(A) and hence y ∈ d_𝒞(A − {y}) ⊆ co(A − {y}) = co(A − {x}). 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_fin(A) such that y ∈ co(F). If x ∉ F, then F ⊆ A − {x}. This means y ∈ co(F − {x}) ⊆ co(A − {x}). If x ∈ F, then by x ∈ d_𝒞(A), there exists G ∈ 𝓟_fin(A) such that x ∈ co(G − {x}). Furthermore, we obtain

y∈co((F−{x})∪{x})⊆co((F−{x})∪co(G−{x}))=co((F−{x})∪(G−{x}))=co((F∪G)−{x})⊆co(A−{x}).

Therefore co(A) = A ∪ d_𝒞(A) ⊆ co(A − {x}).

Sufficiency Assume that x ∈ co(A) and co(A) = co(A − {x}). Then x ∈ co(A − {x}), and hence by Definition 4.1 we have x ∈ d_𝒞(A).□

Theorem 4.9

Let (X, 𝒞) be a convex space and let A be a nonempty set of X. Then A is convexly independent (i.e., x ∉ co(A − {x}) for all x ∈ A) if and only if A ∩ d_𝒞(A) = ∅.

Proof

It is straightforward by Proposition 4.8.

Before proceeding to the next section, we show some examples.

Example 4.10

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

d(A)=co(A)−min(A)∪Max(A),

where min(A) is the minimal element of A and Max(A) is the maximal element of A respectively. As min(A) ⊆ A and Max(A) ⊆ A, it holds that co(A) = A ∪ d(A).
Let X = {a_i ∣ i = 1, 2, 3, 4, 5} be a metric space, and the metric δ on X is defined as figure 1. Note that δ(a₁, a₅) = δ(a₂, a₄) = 3. A subset C of X is called geodesically convex provided

x∈C whenever δ(a,x)+δ(x,b)=δ(a,b) and a,b∈C.

Consider A = {a₁, a₅}, it is trivial to check that co(A) = X, d(A) = X − A and thus co(A) = A ∪ d(A).

$Figure 1 The metric space (X, δ).$
Figure 1
The metric space (X, δ).
Let V be a vector space over a field 𝕂 and let V₀ be the set V minus the zero vector 0. A nonempty set C ⊆ V₀ 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 V₀, and it is trivial to check that for any subset A of V₀,

co(A)=∑i=1nti⋅pi∣p1,p2,⋯pn∈A,t1,t2,⋯tn∈K, and hencex∈d(A) if and only if x=∑i=1nti⋅pi for some p1,p2,⋯pn∈A−{p},t1,t2,⋯tn∈K.

5 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 finite sets). This property entails a new operator, called restricted hull operator, by restricting the hull operator to the family of all finite 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_fin(X) ⟶ 𝓟(X) satisfying the following conditions:

h(∅) = ∅;
for any F ∈ 𝓟_fin(X), F ⊆ h(F);
for any F, G ∈ 𝓟_fin(X), G ⊆ h(F) implies h(G) ⊆ h(F).

It is known that every restricted hull operator can uniquely determine a convex structure [11].

Definition 5.1

A mapping d : 𝓟_fin(X) ⟶ 𝓟(X) is called a restricted c-derived operator provided for any F, G ∈ 𝓟_fin(X), it satisfies the following conditions:

d(∅) = ∅;
F ⊆ G implies d(F) ⊆ d(G);
x ∈ d(F) implies x ∈ d(F − {x});
G ⊆ d(F) implies d(G ∪ F) ⊆ d(F) ∪ F.

Proposition 5.2

Let d : 𝓟_fin(X) ⟶ 𝓟(X) be a restricted c-derived operator. Then the mapping h_d : 𝓟_fin(X) ⟶ 𝓟(X) defined by

∀F∈Pfin(X),hd¯(F):=F∪d¯(F)

is a restricted hull operator.

Proof

The verifications of (H1) and (H2) are trivial. For (H3), Assume that F, G ∈ 𝓟_fin(X) and G ⊆ h_d(F). Let G₁ = G ∩ F and G₂ = G-G₁. Since G ⊆ h_d(F) = d(F) ∪ F, then G₂ ⊆ d(F). By (RD4), we know d(G₂ ∪ F) ⊆ d(F) ∪ F. Furthermore by (RD2), we have

d¯(G)=d¯(G1∪G2)⊆d¯(F∪G2)⊆d¯(F)∪F=hd¯(F).

Hence, h_d(G) = d(G) ∪ G ⊆ h_d(F).□

Lemma 5.3

Every restricted hull operator h is order-preserving, that is, for any F, G ∈ 𝓟_fin(X),

F⊆G⇒h(F)⊆h(G).

Proposition 5.4

Let h be a restricted hull operator on X. Then the mapping d_h : 𝓟_fin(X) ⟶ 𝓟(X) defined by

∀F∈Pfin(X),d¯h(F):={x∣x∈h(F−{x})}

is a restricted c-derived operator.

Proof

By (H1), (H2) and Lemma 5.3, the verifications of (RD1)–(RD3) are straightforward. For (RD4), assume that G ⊆ d_h(F) and x ∈ d_h(G ∪ F). If x ∈ F, then we are done. So assume x ∉ F. Since G ⊆ d(F), we have g ∈ h(F − {g}) for all g ∈ G. It follows that G ⊆ h(F). By (H2) and (H3), F ∪ G ⊆ h(F) and hence h(F ∪ G) ⊆ h(F). Note that x ∉ F, which shows F = F − {x}. Therefore, we have x ∈ h(F ∪ G) ⊆ h(F) = h(F − {x}). This implies x ∈ d_h(F).□

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 suffices to prove (1) d_h_d = d and (2)h_{d_h} = h.

For each F ∈ 𝓟_fin(X), we have

d¯hd¯(F)={x∣x∈hd¯(F−{x})}=x∣x∈F−{x}∪d¯(F−{x})=x∣x∈d¯(F−{x})=x∣x∈d¯(F)=d¯(F).
Take any F ∈ 𝓟_fin(X). On one hand, we have

hd¯h(F)=F∪d¯h(F)={x∣x∈F∪h(F−{x})}⊆h(F).

On the other hand, take any x ∈ h(F). If x ∈ F, then we are done. So assume x ∉ F, and x ∈ h(F) = h(F − {x}) = d_h(F) ⊆ h_{d_h} (F). Hence, h_{d_h} (F) = h(F).□

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_fin(X). Conversely, the c-derived operator of any convex structure on X satisfies the conditions (RD1)-(RD4).

Proof

Let

Cd¯=C∣F∈Pfin(C)⇒d¯(F)⊆C.

It is easy to check 𝒞_d is a convex structure on X. We prove the conclusion in three steps:

We prove co(A) = ⋃{d(F) ∪ F ∣ F ∈ 𝓟_fin(A)} (specially, co(A) = d(A) ∪ A whenever A is finite). Note that {d(F) ∪ F ∣ F ∈ 𝓟_fin(A)} is directed. Then for any finite set G ⊆ ⋃{d(F) ∪ F ∣ F ∈ 𝓟_fin(A)}, there exists H ∈ 𝓟_fin(A) satisfying G ⊆ d(H) ∪ H. It follows that

d¯(G)⊆d¯(d¯(H)∪H)⊆d¯(H)∪H⊆⋃d¯(F)∪F∣F∈Pfin(A),

which means ⋃{d(F) ∪ F ∣ F ∈ 𝓟_fin(A)} ∈ 𝒞_d. Therefore, co(A) ⊆ ⋃{d(F) ∪ F ∣ F ∈ 𝓟_fin(A)}. The converse ⋃{d(F) ∪ F ∣ F ∈ 𝓟_fin(A)} ⊆ co(A) is trivial.
For any F ∈ 𝓟_fin(X), we have

dCd¯(F)={x∈X∣x∈co(F−{x})}=x∈X∣x∈d¯(F−{x})∪(F−{x})=x∈X∣x∈d¯(F−{x})=x∈X∣x∈d¯(F)=d¯(F).
We prove the uniqueness of 𝒞_d. If there exists another convex structure 𝒞^∗ on X with a c-derived operator d^∗ equal to d on 𝓟_fin(X). Then for any A ⊆ X, we have

d∗(A)=⋃{d∗(F)∣F∈Pfin(A)}=⋃d¯(F)∣F∈Pfin(A)=⋃{d(F)∣F∈Pfin(A)}=d(A).

Then by Theorem 4.7, 𝒞_d = 𝒞^∗.□

6 An application to anti-matroids

In this section, we will investigate the relationship between c-derived operators and Shi’s m-derived operators on matroids, and present an application of c-derived operators to anti-maroids.

In a vector space V, let E ∈ 𝓟_fin(V) and let d be a mapping on 𝓟(E) defined by

d(A)=x∈E∣x is the linear combination of A−{x}.

for all A ⊆ E. Xin and Shi [14] generalized this operator as follows.

Let E be a finite set. A mapping d on 𝓟(E) is called a matroid derived operator (m-derived operator for short) [14] if d satisfies the following conditions:

d(∅) = ∅;
A ⊆ B ⇒ d(A) ⊆ d(B);
x ∈ d(A) ⇒ x ∈ d(A − {x});
d(d(A) ∪ A) ⊆ d(A) ∪ A;
y ∈ d(A) − d(A − {x}) ⇒ x ∈ d((A − {x}) ∪ {y}).

Remark 6.1

Since E is a finite set, the Algebraic Law (see (CD4) in Definition 3.1) always holds. That is to say every m-derived operator on a finite set is a c-derived operator.

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 finite set E if and only if it satisfies the following Exchange Law:

(CD5) If p ∉ C ∪ d(C), then p ∈ d(C ∪ {q}) implies q ∈ d(C ∪ {p}).

Proof

Necessity. It suffices to verify (CD5). Since q ∉ C (otherwise p ∈ d(C)), we have C = C ∪ {q} − {q}, implying that p ∈ d(C ∪ {q}) − d((C ∪ {q}) − {q}). Hence by (D5), we obtain q ∈ d((C ∪ {q} − {q}) ∪ {p}) = d(C ∪ {p}).

Sufficiency. It suffices to verify (D5). Take any y ∈ d(A) − d(A − {x}). By (D3), we have y ∈ d(A − {y}) − d(A − {y, x}). Since y ∉ A − {y, x}, by (CD5) we obtain x ∈ d((A − {x, y}) ∪ {y}) = d((A − {x}) ∪ {y}).□

Theorem 6.3

Let (X, 𝒞) be a convex space and let d be the induced c-derived operator. Then d satisfies condition (CD5) if and only if 𝒞 satisfies the Exchange Law:

ifp,q∉co(A),thenp∈co(A∪{q})impliesq∈co(A∪{p}).

Proof

Necessity. Note that co(A) = d(A) ∪ A. It’s trivial if p = q. Now assume p ≠ q. Since p ∈ co(A ∪ {q}) = A ∪ {q} ∪ d(A ∪ {q}) and p ∉ A ∪ {q}, we know p ∈ d(A ∪ {q}). By (CD5), we have q ∈ d(A ∪ {p}) ⊆ co(A ∪ {p}).

Sufficiency. Take any p ∈ d(C ∪ {q}) − (C ∪ d(C)). If q ∈ d(C), then we are done. So assume q ∉ d(C). If q ∈ C, then p ∈ d(C ∪ {q}) = d(C), a contradiction. This shows q ∉ C ∪ d(C). Since p ∈ d(C ∪ {q}) ⊆ co(C ∪ {q}), by the Exchange Law, we obtain q ∈ co(C ∪ {p}) = C ∪ {p} ∪ d(C ∪ {p}). Since q ∉ C and p ≠ q (otherwise by (CD2), p ∈ d(C ∪ {q}) implies p ∈ d(C)), we have q ∈ d(C ∪ {p}).□

Corollary 6.4

Let (X, 𝒞) be a convex space and let d_𝒞 be the induced c-derived operator. Then d_𝒞 satisfies (CD5) if and only if 𝒞 is a matroid.

Theorem 6.5

Let (E, 𝒞) be an anti-matroid. Then co(A) = co(A − d(A)). In particular, co(C − d(C)) = C for all C ∈ 𝒞.

Proof

Take any x ∈ d(A). By Proposition 4.8, we have co(A − {x}) = co(A). Further, by Proposition 2.5 co(A) = co(⋂_x ∈ d(A)}(A − {x})) = co(A − d(A)).□

Proposition 6.6

Let (E, 𝒞) be anti-matroid. Then for any A ⊆ E and C ∈ 𝒞, the following statements hold.

C − d(C) is the generator of C.
A − d(A) is the generator of co(A).
A − d(A) = co(A) − d(co(A)).

Proof

By Theorem 6.5, we know co(C − d(C)) = C. This means that C − d(C) generates C. Furthermore, assume co(B) = C and x ∈ C − d(C). If x ∉ B, then x ∈ d(B) ⊆ d(C), a contradiction. Thus x ∈ B, implying that C − d(C) ⊆ B. Therefore, C − d(C) is the generator of C.
Assume that x ∈ A − d(A) satisfying

co(A)=co((A−d(A))−{x}))=co(A−(d(A)∪{x})).

Note that

x∉(A−(d(A)∪{x}))∪d(A−(d(A)∪{x}))=co(A−(d(A)∪{x}))=co(A),

a contradiction to x ∈ co(A). It follows that co(F)≠ co(A) for all F ⫋ A − d(A). Now assume co(B) = co(A). Note that co(A − d(A)) = co(A) by Theorem 6.5, and thus co((A − d(A))∩ B) = co(A) by Proposition 2.5. It follows that A − d(A) = (A − d(A))∩ B, and hence A − d(A) ⊆ B. Therefore A − d(A) is the generator of co(A).
It’s trivial by the result (1), (2) and Proposition 2.7.□

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 finite set E, then the following statements are equivalent.

(E, 𝒞) is anti-matroid.
(E, 𝒞) is uniquely generated.
For any C ∈ 𝒞, co(C − d(C)) = C.
For any C ∈ 𝒞, C − d(C) is the generator of C.
For any A ∈ 𝓟(E), co(A − d(A)) = co(A).
For any A ∈ 𝓟(E), A − d(A) is the generator of co(A).

Acknowledgement

The authors are extremely grateful to the editor and the referees for their valuable comments and helpful suggestions which help to improve the presentation of this paper. This work is supported by the National Natural Science Foundation of China (11871097) and Joint Ph.D. Program of Beijing Institute of Technology.

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Received: 2018-06-30

Accepted: 2019-02-14

Published Online: 2019-05-11

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0022

Keywords for this article

convex space; derived operator; m-derived operator; matroid; anti-matroid

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