粗糙集参数拟阵
Parametric matroid of rough set
摘要：
粗糙集主要研究对象在论域上通过等价关系的近似。拟阵是向量空间中线性无关性的组合推广。在本文中，我们定义了一个参数集族，它以论域的任意子集为参数，连接粗糙集和拟阵。一方面，对于论域和论域上的等价关系，通过下近似算子定义了一个参数集族。证明了该参数集族满足拟阵的独立集公理，从而可以生成一个拟阵，称为粗糙集的参数拟阵。得到了参数集族的三个等价表示。此外，粗糙集的参数拟阵被证明是一个划分回路拟阵和一个自由拟阵的直和。另一方面，由于划分回路拟阵是通过下近似数得到很好的研究，我们使用它来研究粗糙集的参数拟阵。粗糙集参数拟阵的独立集、基、圈、秩函数和闭包算子等特征用下近似数表示。
英文摘要：
Rough set is mainly concerned with the approximations of objects through an equivalence relation on a universe. Matroid is a combinatorial generalization of linear independence in vector spaces. In this paper, we define a parametric set family, with any subset of a universe as its parameter, to connect rough sets and matroids. On the one hand, for a universe and an equivalence relation on the universe, a parametric set family is defined through the lower approximation operator. This parametric set family is proved to satisfy the independent set axiom of matroids, therefore it can generate a matroid, called a parametric matroid of the rough set. Three equivalent representations of the parametric set family are obtained. Moreover, the parametric matroid of the rough set is proved to be the direct sum of a partition-circuit matroid and a free matroid. On the other hand, since partition-circuit matroids were well studied through the lower approximation number, we use it to investigate the parametric matroid of the rough set. Several characteristics of the parametric matroid of the rough set, such as independent sets, bases, circuits, the rank function and the closure operator, are expressed by the lower approximation number.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
