Editing Matroid (section)

===Graphoids===

[[George J. Minty|Minty]] (1966) defined a ''graphoid'' as a triple <math>(L, C, D)</math> in which <math>C</math> and <math>D</math> are classes of nonempty subsets of <math>L</math> such that 

*(G1) no element of <math>C</math> (called a "circuit") contains another, 

*(G2) no element of <math>D</math> (called a "cocircuit") contains another, 

*(G3) no set in <math>C</math> and set in <math>D</math> intersect in exactly one element, and 

*(G4) whenever <math>L</math> is represented as the disjoint union of subsets <math> R, G, B </math> with <math>G = \{ g \}</math> (a singleton set), then either an <math>X \in C</math> exists such that <math>g \in X \subseteq R \cup G</math> or a <math>Y \in D</math> exists such that <math>g \in Y \subseteq B \cup G</math>.

He proved that there is a matroid for which <math>C</math> is the class of circuits and <math>D</math> is the class of cocircuits. Conversely, if <math>C</math> and <math>D</math> are the circuit and cocircuit classes of a matroid <math>M</math> with ground set <math> E</math>, then <math>(E, C, D)</math> is a graphoid. Thus, graphoids give a ''self-dual cryptomorphic axiomatization'' of matroids.