关于模糊横贯拟阵表示的初步研究

Preliminary Study on the Presentation of Fuzzy Transversal Matroids

本文首先讨论了模糊子集族的全体模糊部分横贯能够形成模糊横贯拟阵的条件,得到两个充要条件。第一个充要条件使用截短模糊子集族来进行描述;第二个充要条件是通过部分横贯的指标集单射兼容这个概念来描述。然后研究了模糊横贯拟阵的表示形式,得到了三个结论。一是任何模糊横贯拟阵都有“精简表示”,而且构造了从一个表示寻找精简表示的方法;二是模糊横贯拟阵的表示的模糊子集个数不小于导出拟阵的最大秩;三是一致模糊横贯拟阵的任何表示都包含“最小表示”,也给出了求这个“最小表示”的方法。

The paper has firstly discussed the condition which all fuzzy partical transversal of a fuzzy set family is a fuzzy transversal matroid.First sufficient and necessary condition has been described by the truncations of family of fuzzy sets.Another sufficient and necessary condition has been researched by the compatibility of injective mappings of partical transversal index sets.Then,this paper has studied representations of fuzzy transversal matroids,gotten three conclusions.It is first conclusion that any fuzzy transversal matroids have“simplified representations”.The methord of constructing simplified representations has also been provided;Second conclusion is that the number of fuzzy sets in one presentation is not less than the maximum rank of induced matroids;Third conclusion is that any uniform fuzzy transversal matroids have“minimum representations”.The way which finding out the minimum representation from a representation of uniform fuzzy transversal matroids has been given.