Oriented matroid

An oriented matroid is a <a href="/facts/Mathematics/pxTouaz4">mathematical</a> <a href="/facts/Mathematical_structure/6CXReUNv">structure</a> that abstracts the properties of <a href="/facts/Directed_graph/YBdfQ0um">directed graphs</a>, <a href="/facts/Vector_space/M1pxMLx2">vector</a> arrangements over ordered fields, and <a href="/facts/Arrangement_of_hyperplanes/78c23rYy">hyperplane arrangements</a> over <a href="/facts/Ordered_field/an0prSgG">ordered fields</a>. In comparison, an ordinary (i.e., non-oriented) <a href="/facts/Matroid/5bLcGlRF">matroid</a> abstracts the <a href="/facts/Linear_independence/8JrHfIIa">dependence</a> properties that are common both to <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graphs</a>, which are not necessarily directed, and to arrangements of vectors over <a href="/facts/Field_(mathematics)/xAjAS4ko">fields</a>, which are not necessarily ordered.

All oriented matroids have an underlying <a href="/facts/Matroid/5bLcGlRF">matroid</a>. Thus, results on ordinary matroids can be applied to oriented matroids. However, the <a href="/facts/Conversion_(logic)/VhlUrxxx">converse</a> 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 <a href="/facts/Dimension_theory_(algebra)/qDqrj97u">dimension theory</a> and <a href="/facts/Algorithms/fnl5NmRt">algorithms</a>.
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 <a href="/facts/Geometry/wTQf5ffQ">geometry</a> and <a href="/facts/Optimization_(mathematics)/oRn8Iv5I">optimization</a>.

Oriented matroid open-in-new

Oriented matroid