NoteOrientable arithmetic matroids
Section snippets
Definitions
Let be a finite totally ordered set. We will frequently make use of -tuples of elements of , so with an abuse of notation for any set we will write for the increasing tuple .
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 is a non-empty set such that
Since this definition
Deletion
The deletion of 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 satisfies the first two conditions of Definition 1.6.
Proposition 2.1 The triple is an oriented arithmetic matroid.
Proof Let be the rank of and such that . Consider the elements and in . For
Contraction
The contraction (or restriction) of 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 be a subset of and call its rank. We choose an independent list of elements in . Define as and .
Proposition 3.1 The triple is an oriented arithmetic matroid.
Proof We call
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 is ordered. For every we call the complement of in with some arbitrary order and let be the sign of the permutation that reorders the list as they appear in . We define as 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 restricted to a finite (multi-)set .
Definition 5.1 A map is a GP-function if it is alternating and for all and all the following equality holds
We denote the point-wise product of two function and with
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 and be two oriented arithmetic matroids over the same matroid . Then is a re-orientation of .
We fix a total order on such that , the first 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 of a matroid is the graph on the set of vertices with an edge between two vertices and if .
Once chosen a basis of a
Representability
Definition 7.1 An arithmetic matroid has the strong GCD property if for all .
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 and are GCD arithmetic matroids.
Proposition 7.2 Let be an orientable arithmetic matroid. Then the underlying matroid is representable over .
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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