极小圈模对与二元域拟阵的特征

Circuit Modular Pair and Characterizations of Binary Matroids

研究极小圈模对与二元域拟阵的特征.首先给出拟阵M的极小圈模对,模对的并的秩与相应的超平面的交的秩三者的等价关系.在两个极小圈不等的条件下,证明了满足极小圈消去公理的极小圈是唯一的并且极小圈模对的对称差包含在其中,结合极小圈的对称差的表示,证明了极小圈与基的差的绝对值大于等于2.后面两个证明都把原来的必要条件推广为充要条件.最后,用M上不相同的极小圈,极小圈模对,极小圈的对称差表示,M上不相等的超平面,超平面的并不等于E及满足的秩等式极简单地刻划了二元域拟阵M的特征.

The properties of circuit modular pair and characterizations of binary matroids are investigated.First,the equivalence relations among the circuit modular pair over matroid M,the rank of the union of modular pair and the rank of the intersection of the corresponding hyperplane are studied.Under the condition that two circuits are not equal,the uniqueness of circuit satisfying circuit eUmination and the symmetric difference of circuit modular pahis a subset of a circuit over Mare proved,combining the representation of the symmetric difference of circuits,the fact that the absolute value of the difference of circuit and the basis for M is greater or equal to 2 is given.In the two proofs of above,the necessary conditions in[1]of circuit modular pair are showed to be sufficient conditions of circuit modular pair.Finally,some characterizations of binary matroids are easily deduced based on two different circuit over M,the representation of symmetric difference,two different hyperplane over M,the union of hyperplanes which satisfying the rank equation is not equal to E.