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In matroid theory, a binary matroid is a matroid that can be represented over the finite field GF(2).〔.〕 That is, up to isomorphism, they are the matroids whose elements are the columns of a (0,1)-matrix and whose sets of elements are independent if and only if the corresponding columns are linearly independent in GF(2). ==Alternative characterizations== A matroid is binary if and only if *It is the matroid defined from a symmetric (0,1)-matrix.〔.〕 *For every set of circuits of the matroid, the symmetric difference of the circuits in can be represented as a disjoint union of circuits.〔. Reprinted in , pp. 55–79.〕〔, Theorem 10.1.3, p. 162.〕 *For every pair of circuits of the matroid, their symmetric difference contains another circuit.〔 *For every pair where is a circuit of and is a circuit of the dual matroid of , is an even number.〔〔.〕 *For every pair where is a basis of and is a circuit of , is the symmetric difference of the fundamental circuits induced in by the elements of .〔〔 *No matroid minor of is the uniform matroid , the four-point line.〔.〕〔.〕〔, Section 10.2, "An excluded minor criterion for a matroid to be binary", pp. 167–169.〕 *In the geometric lattice associated to the matroid, every interval of height two has at most five elements.〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「binary matroid」の詳細全文を読む スポンサード リンク
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