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Open Mathematics

formerly Central European Journal of Mathematics

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Volume 14, Issue 1 (Jan 2016)

Issues

Factorization theorems for strong maps between matroids of arbitrary cardinality

Hua Mao
  • Corresponding author
  • Department of Mathematics, Hebei University, Baoding 071002, China
  • Email:
Published Online: 2016-10-10 | DOI: https://doi.org/10.1515/math-2016-0060

Abstract

In this paper we present factorization theorems for strong maps between matroids of arbitrary cardinality. Moreover, we present a new way to prove the factorization theorem for strong maps between finite matroids.

Keywords: Strong map; Factorization; Matroid; Arbitrary cardinality

MSC 2010: 05B35; 03E05

1 Introduction and preliminaries

Much attention has been given to strong maps in the category of finite matroids (cf. [15], [6, Ch.7], [17, Ch.8]). The most well known results for strong maps in the category of finite matroids are factorization theorems.

Matroid theory is a theory not only of finite matroids but also of infinite matroids. If the factorization theorems do not extend to the case of infinite matroids they would appear to be incomplete. Therefore we need to find the factorization theorems of matroid theory for the class of infinite matroids.

Oxley points out in [8] that there is no single class of structures given the name infinite matroids. A variety of classes of matroid-like structures on infinite sets have been studied for different reasons by various authors.

Many of the recent results in infinite matroid theory are for matroids of arbitrary cardinality (cf. [915]). Matroids of arbitrary cardinality seem therefore to be the most studied classes of infinite matroids yielding fruitful results. In this paper we will adopt the definition of infinite matroid given in [9], i.e. the definition of matroid of arbitrary cardinality, and study the factorization theorems for strong maps of infinitie matroids of this type.

We start by reviewing those aspects of matroid theory which we will need. Firstly, we assume that E is some arbitrary, possibly infinite set; PE is the set of all subsets of E; ℕ0 = {0, 1, 2...}.

Definition 1.1 ([9]):

  • (1)

    Assume m ∈ ℕ0 and FPE. Then the pair M:=E, F is called a matroid of rank m with F as its closed sets, if the following axioms hold:

    • (F1)

      EF;

    • (F2)

      If F1, F2F, then F1F2F;

    • (F3)

      Assume F0F and x1, x2E \ F0. Then one has either:

      • (i)

        FF|F0x1F=FF|F0x2F or

      • (ii)

        F1F2 = F0 for certain F1, F2F containing F0 ∪ {x1} or F0 ∪ {x2}, respectively;

    • (F4)

      M = max{n ∈ ℕ0|there exist F0, F1, , FnF. with F0F1 ⊂ ⋯ ⊂ Fn = E}.

  • (2)

    Let M=E, Fbe a matroid. The closure operator σ=σM:P(E)F of M is defined by σA:=AFFF F. M is called simple, if any subset AE with |A| ≤ 1 lies in F.

Remark: The definition of a finite matroid can be found in [5, Ch.1], [6, Ch.1] and [7, Ch.2] that of identity map is cf. [16, p.12] and rankfunction of a matroid of arbitrary cardinality is come from [9]. From these definitions and their some relative properties, it is easy to know that any finite matroid on E is a matroid of arbitrary cardinality on E. Hence, in this paper except explaining in a special way a matroid always means a matroid of arbitrary cardinality defined in Definition 1.1.

Lemma 1.2 ([9]): Assume M=E, Fis a matroid with σ as its closure operator Then for any family (Fi)iI of closed sets in M, one has also F:=iIFiF. σ(A) is the smallest set in F containing AE. In particular, one has σA=AAF. Moreover A ⊆ σ(A) = σ(σ(A)) holds for AE; for AE; for ABE, one has σ(A) ⊆ σ(B).In the rest of this section we give a few results and properties of matroids which will be needed in the next section.

Lemma 1.3:

  • (1)

    Let M1=E1, F1, M2=E2, F2 be matroids on disjoint sets E1, E2 with m1, m2 as their ranks, respectively Let F=F1F2|F1F1, F2F2. Then M=E=E1E2, F is a matroid defined on E with rank m1 + m2. We call M the direct sum of M1 and M2, written M1 + M2.

  • (2)

    Let M=E, Fbe a matroid with σ as its closure operator TE and FT=σAT|AT. Then T, FT is a matroid on T. We denote this matroid by M|T and call it the restriction of M to T.

The following definitions are for matroids of arbitrary cardinality. They are simply generalizations of the corresponding definitions for finite matroids (cf. [6, 7]).

Definition 1.4:

  • (1)

    Let M=E, Fbe a matroid with σ as its closure operator Then

    A loop of M is an element x of E such that x ∈ σ(ø).

    If x, yE and xy, then x is parallel to y if and only if x ∈ σ(y) and y ∈ σ(x) and neither x nor y are loops.

    Let TE and N=T, F be a matroid with σ′ as its closure operator satisfying the following statement: if for any AT and xT \ A, it is always true that x ∈ σ′(A) ⇔ x ∈ σ (A ∪ (E \ T)), then N will be called the contraction of M to T denoted by M · T.

  • (2)

    Two matroids M1 and M2 on E2 and E2 are isomorphic, denoted by M1M2, if there is a bijection φ : E1E2 which preserves the closure operation.

  • (3)

    Let M be a matroid on E and 0 be an element not in E. The matroid M0 is the direct sum M + {0} on the set E ∪ {0}, where {0} is the matroid of rank zero on the single-element set {0}. We use the same symbol 0 for the zero of any matroid M0.

    Let M=(E, FM) and N=T, FN be matroids. A strong map f from M to N is a function f : E ∪ {0} → T ∪ {0}, mapping 0 to 0, and satisying the following condition: the inverse image of any closed set of N0 is a closed set of M0.

Note. We use f to denote both the function E ∪ {0} → T ∪ {0}, and its restriction to E. Also by a map ET we will mean a map g : E ∪ {0} → T ∪ {0} in which g(0) = 0.By Lemma 1.3 and Definition 1.4, it is easy to obtain the following Lemma 1.5 which is a result about a type of strong map which arises from “submatroids”. We will see in the following section that these maps play an important role in factorization theorems for strong maps.

Lemma 1.5:

  • (1)

    Let Mi be a matroid on Ei(i = 1, 2) where E1E2 = ø. Then Mi = (M1 + M2)|Ei, Mi = (M1 + M2Ei, (i = 1, 2).

  • (2)

    Let Mi be a matroid on Ei(i = 1, 2, 3) and g : M1M2 and f : M2M3 be strong maps. Then fg : M1M3 is a strong map.

  • (3)

    Let M=E, F be a matroid. Then for TE and N=M|T, the inclusion map i : T → E defines a strong map, the injection map. For SE and N=M·S, define c : ES by cx=x, xS0, xS contraction map c is a strong map. Let s(M) be the simplification of M, that is all mutually parallel elements of M are identified in s(M) and all loops of M are sent to the zero element in s(M). Any simplification is a strong map. Specifically when M · T exists, we define the projection of M to TE as the composite map sc : Ms(M · T). Then the projection map is a strong map.

2 Factorization theorems

In this section we discuss factorization theorems for strong maps between matroids and present the two main results of this paper, Theorem 2.1 and Theorem 2.3.

To begin we note that the result of [6, p. 315, Theorem 3] is that of Crapo [1] and Higgs [2]. In [4], Mao and Liu have pointed out that [2, Theorem III] is wrong and as noted in [4] the discussion in Crapo is correct but the definition of strong map in Crapo is only a special case of that in [6]. Therefore [1] and [2] are of no use to the present discussion.

Other references to factorization theorems for strong maps are [3, 8.2.7 and 8.2.8], [5, pp. 267-268, Proposition 7.3.11, Corollary 7.3.12 and Corollary 7.3.13], [6, Ch.17, § 4]. But the content of [3, 5] is the same as that of [6, Ch.17, § 1] which in turn is a special case of [6, pp. 315-316, Theorem 3 and Theorem 4]. In other words it is not useful to revise the contents of [6, pp. 315-316, Theorem 3 and Theorem 4], [3, 5] and [6, Ch.17, § 4].

We will see that Theorem 2.1 is simply a generalization of the corresponding result for finite matroids (cf. [6, p.313, Theorem 1]) In order to draw a comparison between the proof of our Theorem 2.1 and that of [6, p.313, Theorem 1] we mimic the structure of the proof in [6, p.313, Theorem 1]. Of course the mathematical content differs.

Theorem 2.1: Let M1=E1, F1 and M2=E2, F2 be matroids on disjoint sets E1, E2 and suppose there exist strong maps fi : MiM (i = 1, 2), where M=(E, FM) is a matroid on E. Then there is a unique map f:(M1+M2=(E1E2;FM1+M2))M such that the figure below commutes where i1 and i2 are the inclusion maps.

Proof: Define f : E1E2E by f(x)={f1(x), xE1f2(x), xE2. For any FFM , by the strong property of fi one has fi1(F)Fi(i=1;2), and further, f1(F)=f11(F)f21(F)FM1+M2 according to Lemma 1.3. Hence f is strong. Evidently, f1=fi1 and f2=fi2. The uniqueness of f is easy to see.

Before presenting the second main result of this section, Theorem 2.3, we consider the following preparatory results.

Lemma 2.2: Let f:M=(E, FM)N=(T, FN) be a surjective and strong map and let N be a simple map. Then f is (up to isomorphism) a projection.

Proof: Let F1={f1(B)|BFN} and M1=(E, F1) . Then it is easy to check that M1 satisfies (F1)-(F4). So, M1 is a matroid.Let σM1 be the closure operator of M1. For any xE, if f(x)=a, then one has xσM1(x)f1(a) in view of Lemma 1.2. σM1(x)F1 implies that there is BFN satisfying f−1(B)=σM1(x), and so f(x)∈B⊆{a}. Hence B={a} holds. Furthermore, it follows that σM1(x)=f−1(a). Since N is simple, one that if x is aloop of M1, then xσM1(∅)=σM1(0)=f−1(0). Conversely, suppose 0≠yf−1(0). One has σM1(y)⊆f−1(0)=σM1(0)=σM1(∅), and so y is aloop of M1. That is f−1(0)={xE|x is aloop of M1} ∪{0} is true. In addition, if x is parallel to y in M1, i.e xσM1(y), yσM1(x) and x, yf−1(0), we must have f(x)=f(y). We notice that if yf−1(a) where a=f(x), then f(y)=a, and furthermore σM1(y)=f−1(a)=σM1(x). This implies that x is parallel to y.Denote f–1(0) by K and set F2={XE\K|XKF1}. Then it is straightforward to show that (F1)-(F4) are verified by (E\K, F2) . Let σ2 be the closure operator of (E\K, F2) . One obtains σ2(A)KF1 holds for ∀AE\K, that is there exists BFN such that σ2(A)∪K=f−1(B)=f−1(B)∪K=σM1(AK). Let x∈(E\K)\A=E\(KA). Then xσ2(A) if and only if xσM1(AK)=σM1(A∪(E\(E\K))). Consequently, (E\K, F2)=M1(E\K)=c(M1) holds true.Let s(c(M1))=(ES, Fs) be the simplification of c(M1). By the properties of c(M1) and M1, one obtains that Es={a1|aT, s(c(f1(a)))=a1} . Let g:(ES, Fs)N be defined by g: a−1a. Clearly g is a bijection. For A={aαT|αA}FN , one has f1(A)={f1(aα)|αA}K by the property of f. Evidently K{f1(aα)|αA}= , and so {f1(aα)|αA}F2 . Further s({f1(aα)|αA})={aα1|αA}Fs , i.e. g1(A)Fs . Similarly, by the properties and definitions of M1c(M1) and s(c(M1)), one obtains that for {aα1|αA}Fs , we have A={aαT|αA}FN . Hence g is an isomorphism.Let id be the identity map on E ∪ {0}, from M to M1. It is easy to see that id is a strong map. Therefore (gsc(id)) is astrong map from M to N. Furthermore, for every xE, if f(x) = aT then (gsc(id))(x)= (gsc)(id(x))=(gsc)(x). We see that when a = 0 one has xK and so c(x) = 0, and furthermore s(c(x)) = 0, g(s(c(x))) = 0 = f(x); when a ≠ 0, one obtains xE\K, i.e. xf–1(a), and so c(x) = x, and furthermore s(c(x))=a–1, g(s(c(x)))=g(a–1) = a = f(x).Consequently, f(x)=g(s(c(id(x)))) = g(s(c(x))) = g((sc)(x)) for xE. Hence f is (up to isomorphism) a projection sc.

Fig. 1

Using Lemma 2.2, we obtain Theorem 2.3.

Theorem 2.3: Let 5 M=(E, FM) and N=T, FN be matroids. Let N furthermore be simple. If f : MN is a strong map, then f is (up to isomorphism) the composition of an injection and a projection.

Proof: Let h : TT′ be a bijection and T′∩E=∅=T′∩T, FN={h(F)T|h(F)={h(x)|xF} , where FFN} . Clearly, N=(T, FN) is a matroid on T′ and NN′. Here, up to isomorphism, f is a strong map from M to N′. From this we can assume that M and N are on disjoint sets E and T.Let j, j′ be the natural injections, i.e. j : MM + N and j′ : NM + N. Let id be the identity function id : NN. By Theorem 2.1, there is a strong map g : M + NN such that the diagram below commutes and g(x)={f(x), xEx, xT . By Lemma 1.5, N = (M + N) ⋅ T holds. Clearly TET and g : M + NN is surjective. Using Lemma 2.2 we deduce that up to isomorphism, g is a projection, so that f is (up to isomorphism) the composition of an injection and a projection.

Fig. 2

A further result for this section is based on the following observation. By [4] it is evident that the proof of [6, p.315, Theorem 3] is incorrect. The proof of the important result [6, p.316, Theorem 4] relies on [6, p.315, Theorem 3]. This makes us question the validity of [6, p.316, Theorem 4]. By Theorem 2.3 and the property that any finite matroid is a matroid of arbitrary cardinality it is evident that [6, p.316, Theorem 4] should read as follows:

“Let f : MN be a strong map where M, N are finite matroids and N is simple. Then f is (up to isomorphism) the composition of an injection and a projection.”

So in the finite case the method presented here differs from that of Welsh in [6].

In this paper we used the definition of infinite matroid given in [9]. Comparing the definition of infinite matroid given in [9] with that in [17] (i.e. finitary matroids in [17]), we find that the definition in [17] is more general than that in [9]. In addition, the authors of [17] not only compared the properties of finite matroids with those of ‘B-matroids’ by Oxley [8], but also sought to find a formulation of the axioms of infinite matroids. Hence, in future work we may consider the factorization theorems for strong maps between infinite matroids which satisfy the axioms of a finite matroid given in [17].

Acknowledgement

This paper is granted by the National Nature Science Foundation of China (61572011).

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About the article

Received: 2015-04-11

Accepted: 2016-08-25

Published Online: 2016-10-10

Published in Print: 2016-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0060. Export Citation

© 2016 Mao, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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