# Oriented matroid Oriented-matroid theory allows a combinatorial approach to the max-flow min-cut theorem. A network with the value of flow equal to the capacity of an s-t cut

An oriented matroid is a mathematical structure that abstracts the properties of directed graphs and of arrangements of vectors in a vector space over an ordered field (particularly for partially ordered vector spaces). In comparison, an ordinary (i.e., non-oriented) matroid abstracts the dependence properties that are common both to graphs, which are not necessarily directed, and to arrangements of vectors over fields, which are not necessarily ordered. 

All oriented matroids have an underlying matroid. Thus, results on ordinary matroids can be applied to oriented matroids. However, the converse is false; some matroids cannot become an oriented matroid by orienting an underlying structure (e.g., circuits or independent sets). The distinction between matroids and oriented matroids is discussed further below.

Matroids are often useful in areas such as dimension theory and algorithms. Because of an oriented matroid's inclusion of additional details about the oriented nature of a structure, its usefulness extends further into several areas including geometry and optimization.

## Background

In order to abstract the concept of orientation on the edges of a graph to sets, one needs the ability to assign "direction" to the elements of a set. The way this achieved is with the following definition of signed sets.

The members of $X^{+}$ are called the positive elements; members of $X^{-}$ are the negative elements.

Given an element of the support $x$ , we will write $x$ for a positive element and $-x$ for a negative element. In this way, a signed set is just adding negative signs to distinguish elements. This will make sense as a "direction" only when we consider orientations of larger structures. Then the sign of each element will encode its direction relative to this orientation.

## Axiomatizations

Like ordinary matroids, several equivalent systems of axioms exist. (Such structures that possess multiple equivalent axiomatizations are called cryptomorphic.)

### Chirotope axioms

The term chirotope is derived from the mathematical notion of chirality, which is a concept abstracted from chemistry used to distinguish molecules.

### Equivalence

$C^{+}=\{x_{i}:(-1)^{i}\chi (x_{1},\dots ,x_{i-1},\dots ,x_{i+1},\dots ,x_{r+1})=1\}$ and negative elements the compliment. Thus a chirotope gives rise to the oriented bases of an oriented matroid. In this sense, (B0) is the nonempty axiom for bases and (B2) is the basis exchange property.

## Examples

Oriented matroids are often introduced (e.g., Bachem and Kern) as an abstraction for directed graphs or systems of linear inequalities. Below are the explicit constructions.

### Directed graphs

{{#invoke:main|main}} {{#invoke:see also|seealso}} Given a digraph, we define a signed circuit from the standard circuit of the graph by the following method. The support of the signed circuit $\textstyle {\underline {X}}$ is the standard set of edges in a minimal cycle. We go along the cycle in the clockwise or anticlockwise direction assigning those edges whose orientation agrees with the direction to the positive elements $\textstyle X^{+}$ and those edges whose orientation disagrees with the direction to the negative elements $\textstyle X^{-}$ . If $\textstyle {\mathcal {C}}$ is the set of all such $\textstyle X$ , then $\textstyle {\mathcal {C}}$ is the set of signed circuits of an oriented matroid on the set of edges of the directed graph.

### Linear algebra

{{#invoke:see also|seealso}} If $\textstyle E$ is any finite subset of $\textstyle {\mathbb {R} }^{n}$ , then the set of minimal linearly dependent sets forms the circuit set of a matroid on $\textstyle E$ . To extend this construction to oriented matroids, for each circuit $\textstyle \{v_{1},\dots ,v_{m}\}$ there is a minimal linear dependence

$\sum _{i=1}^{m}\lambda _{i}v_{i}=0$ $\chi (x_{1},\dots ,x_{r})={\text{sign}}(\det(x_{1},\dots ,x_{r}))$ where the right hand side of the equation is the sign of the determinant. Then $\chi$ is the chirotope of the same oriented matroid on the set $E$ .

### Convex polytope

{{#invoke:main|main}} Ziegler introduces oriented matroids via convex polytopes.

## Results

### Orientability

A standard matroid is called orientable if its circuits are the supports of signed circuits of some oriented matroid. It is known that all real representable matroids are orientable. It is also known that the class of orientable matroids is closed under taking minors, however the list of forbidden minors for orientable matroids is known to be infinite. In this sense, oriented matroids is a much stricter formalization than regular matroids.

### Duality

{{#invoke:see also|seealso}} Much like matroids have unique dual, oriented matroids have unique orthogonal dual. What this means is the underlying matroids are dual and that the cocircuits are signed so that they are orthogonal to every circuit. Two signed sets are said to be orthogonal if the intersection of their supports is empty or if the restriction of their positive elements to the intersection and negative elements to the intersection form two nonidentical and non-opposite signed sets. The existence and uniqueness of the dual oriented matroid depends on the fact that every signed circuit is orthogonal to every signed cocircuit. To see why orthogonality is necessary for uniqueness one needs only to look to the digraph example above. We know that for planar graphs, that the dual of the circuit matroid is the circuit matroid of the graph's planar dual. Thus there are as many different oriented matroids that are dual as there are ways to orient a graph and its dual.

To see the explicit construction of this unique orthogonal dual oriented matroid, consider an oriented matroid's chirotope $\chi :E^{r}\rightarrow \{-1,0,1\}$ . If we consider a list of elements of $x_{1},\dots ,x_{k}\in E$ as a cyclic permutation then we define ${\text{sign}}(x_{1},\dots ,x_{k})$ to be the sign of the associated permutation. If $\chi ^{*}:E^{|E|-r}\rightarrow \{-1,0,1\}$ is defined as

$\chi ^{*}(x_{1},\dots ,x_{r})\mapsto \chi (x_{r+1},\dots ,x_{|E|}){\text{sign}}(x_{1},\dots ,x_{r},x_{r+1},\dots ,x_{|E|}),$ then $\chi ^{*}$ is the chirotope of the unique orthogonal dual oriented matroid.

### Topological representation

Oriented matroids are abstractions of geometric constructs. The exact constructs are arrangements of pseudospheres. A $d$ -dimensional pseudosphere is an embedding of $e:S^{d}\hookrightarrow S^{d+1}$ such that there exists a homeomorphism $h:S^{d+1}\rightarrow S^{d+1}$ so that $h\circ e$ embeds $S^{d}$ as an equator of $S^{d+1}$ . In this sense a pseudosphere is just a tame sphere (as opposed to wild spheres). A pseudosphere arrangement in $S^{d}$ is a collection of pseudospheres that intersect along pseudospheres. The Folkman Lawrence topological representation theorem states that every oriented matroid of rank $d+1$ can be obtained from a pseudosphere arrangement in $S^{d}$ .

### Geometry A zonotope, which is a Minkowski sum of line segments, is a fundamental model for oriented matroids. The sixteen dark-red points (on the right) form the Minkowski sum of the four non-convex sets (on the left), each of which consists of a pair of red points. Their convex hulls (shaded pink) contain plus-signs (+): The right plus-sign is the sum of the left plus-signs.

{{#invoke:main|main}} {{#invoke:see also|seealso}} The theory of oriented matroids has influenced the development of combinatorial geometry, especially the theory of convex polytopes, zonotopes, and of configurations of vectors (arrangements of hyperplanes). Many results—Carathéodory's theorem, Helly's theorem, Radon's theorem, the Hahn–Banach theorem, the Krein–Milman theorem, the lemma of Farkas—can be formulated using appropriate oriented matroids.

### Optimization In convex geometry, the simplex algorithm for linear programming is interpreted as tracing a path along the vertices of a convex polyhedron. Oriented matroid theory studies the combinatorial invariants that are revealed in the sign-patterns of the matrices that appear as pivoting algorithms exchange bases.