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Orientable arithmetic matroids

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Abstract

The theory of matroids has been generalized to oriented matroids and, recently, to arithmetic matroids. We want to give a definition of “oriented arithmetic matroid” and prove some properties like the “uniqueness of orientation”.

Section snippets

Definitions

Let E be a finite totally ordered set. We will frequently make use of r-tuples of elements of E, so with an abuse of notation for any set A={a1,,ar}E we will write A for the increasing tuple (a1,,ar).

We give the definition of a matroid in terms of its basis, since [19, Theorem 1.2.3] shows that it is equivalent to the one given in terms of independent sets.

Definition 1.1

A matroid over a finite set E is a non-empty set BP(E) such that B1,B2BxB1B2yB2B1 such that B1{x}{y}B.

Since this definition

Deletion

The deletion of AE is an operation defined for matroids [19, p. 22], for oriented matroids [2, p. 133], and for arithmetic matroids [5, section 4.3] [3, section 3]. We now define a deletion operation for oriented arithmetic matroids.

The triple (EA,χA,mA) satisfies the first two conditions of Definition 1.6.

Proposition 2.1

The triple (EA,χA,mA) is an oriented arithmetic matroid.

Proof

Let s be the rank of EA and f̲=(a1,,ars)A such that rk((EA)f̲)=r. Consider the elements x2,,xs and y0,,ys in EA. For

Contraction

The contraction (or restriction) of AE is an operation defined for matroids [19, p. 22], for oriented matroids [2, p. 134], and for arithmetic matroids [5, section 4.3] [3, section 3]. We now define a contraction operation for oriented arithmetic matroids.

Let A be a subset of E and call rs its rank. We choose an independent list f̲=(a1,,ars) of elements in A. Define χA:(EA)s{1,0,1} as χA(z̲)=χ(z̲f̲) and mA(S)=m(AS).

Proposition 3.1

The triple (EA,χA,mA) is an oriented arithmetic matroid.

Proof

We call T=

Duality

The duality is an operation defined for matroids [19, chapter 2], for oriented matroids [2, p. 135], and for arithmetic matroids [5, p. 339] [3, p. 5526]. We now define duality for oriented arithmetic matroids.

Recall that the set E is ordered. For every z̲=(z1,,zk)E we call z̲ the complement of z̲ in E with some arbitrary order and let σ(z̲,z̲) be the sign of the permutation that reorders the list (z̲,z̲) as they appear in E. We define χ:Enr{1,0,1} as χ(z̲)=χ(z̲)σ(z̲,z̲)and the

GP-functions

We now study functions satisfying a relation that looks like the Plücker relation for the Grassmannian. A posteriori all these functions are nothing else that the determinant det:VrQ restricted to a finite (multi-)set EV.

Definition 5.1

A map f:ErQ is a GP-function if it is alternating and for all x̲Er1 and all y̲Er+1 the following equality holds i=0r(1)if(yi,x2,xr)f(y0,yi1,yi+1yr)=0.

We denote the point-wise product of two function χ and m with χm(b̲)=defχ(b̲)m(b̲).

Example 5.2

The main examples of GP-function

Uniqueness of the orientation

This section is dedicated to prove the following theorem:

Theorem 6.1

Let (E,χ,m) and (E,χ,m) be two oriented arithmetic matroids over the same matroid (E,rk). Then χ is a re-orientation of χ.

We fix a total order on E[n] such that [r], the first r elements, are a basis of the matroid.

The basis graph of a matroid was first studied in [16] and [17].

Definition 6.2

The basis graph BG of a matroid (E,B) is the graph on the set B of vertices with an edge between two vertices B1 and B2 if |B1B2|=1.

Once chosen a basis B0 of a

Representability

Definition 7.1

An arithmetic matroid (E,rk,m) has the strong GCD property if m(A)=gcd{m(B)B basis and |BA|=rkA}for all AE.

Notice that arithmetic matroids with the strong GCD property are uniquely determined by their values on the basis of the underlying matroid. The strong GCD property is equivalent to the statement that both (E,rk,m) and (E,rk,m) are GCD arithmetic matroids.

Proposition 7.2

Let (E,rk,m) be an orientable arithmetic matroid. Then the underlying matroid (E,rk) is representable over Q.

Proof

We choose an

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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