Factorization theorems for strong maps between matroids of arbitrary cardinality

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.

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. [9][10][11][12][13][14][15]). 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; P.E/ is the set of all subsets of E; N 0 D f0; 1; 2; : : :g.
(1) Assume m 2 N 0 and F Â P.E/. Then the pair M WD .E; F/ is called a matroid of rank m with F as its closed sets, if the following axioms hold: Then one has either: (2) Let M D .E; F/ be a matroid. The closure operator 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 rank function 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. 9]). Assume M D .E; F/ is a matroid with as its closure operator. Then for any family .F i / i 2I of closed sets in M , one has also F WD T In the rest of this section we give a few results and properties of matroids which will be needed in the next section. 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 D .E; F/ be a matroid with as its closure operator. Then A loop of M is an element x of E such that x 2 .;/. If x; y 2 E and x ¤ y, then x is parallel to y if and only if x 2 .y/ and y 2 .x/ and neither x nor y are loops. Let T Â E and N D .T; F 0 / be a matroid with 0 as its closure operator satisfying the following statement: if for any A Â T and x 2 T n A, it is always true that Note. We use f to denote both the function E [ f0g ! T [ f0g and its restriction to E. Also by a map g W E ! T we will mean a map g W E [ f0g ! T [ f0g in which g.0/ D 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.

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 [ We will see that Theorem 2.1 is simply a generalization of the corresponding result for finite matroids (cf. Proof  Let id be the identity map on E [ f0g from M to M 1 . It is easy to see that id is a strong map. Therefore .gsc.id // is a strong map from M to N . Furthermore, for every x 2 E, if f .x/ D a 2 T then .gsc.id //.x/ D .gsc/.id.x// D .gsc/.x/. We see that when a D 0 one has x 2 K and so c.x/ D 0, and furthermore s.c.x// D 0; g.s.c.x/// D 0 D f .x/; when a ¤ 0, one obtains x 2 E nK, i.e. x 2 f 1 .a/, and so c.x/ D x, and furthermore s.c.x// D a 1 ; g.s.c.x/// D g.a 1 / D a D f .x/. Consequently, f .x/ D g.s.c.id.x//// D g.s.c.x/// D g..sc/.x// for x 2 E. Hence f is (up to isomorphism) a projection sc.
Using Lemma 2.2, we obtain Theorem 2.3. Proof. Let h W T ! T 0 be a bijection and T 0 \ E D ; D T 0 \ T; F N 0 D fh.F / Â T 0 jh.F / D fh.x/jx 2 F g; where F 2 F N g. Clearly, N 0 D .T 0 ; F N 0 / is a matroid on T 0 and N ' N 0 . Here, up to isomorphism, f is a strong map from M to N 0 . From this we can assume that M and N are on disjoint sets E and T .
Let j; j 0 be the natural injections, i.e. j W M ! M C N and j 0 W N ! M C N . Let id be the identity function id W N ! N: By Theorem 2.1, there is a strong map g W M C N ! N such that the diagram below commutes and 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.
A further result for this section is based on the following observation. By [4] it is evident that the proof of [ "Let f W M ! N 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].