关系近似数及其诱导的拟阵结构

Relational Approximation Number and the Induced Matroidal Structure

二元关系在数学中是一种非常重要的结构,并且这种结构已经被作为一些领域的基础.拟阵论是线性代数理论和图论的推广,其具有完善的理论体系且被广泛应用到许多领域.类比上近似数,通过二元关系提出了关系近似数的概念.证明了关系近似数满足次模性,同时通过计算前继邻域的基数的方法给出了关系近似数的计算方法.此外,通过引入多重集族的概念,给出了一个集合的上近似数和关系近似数相等的充分必要条件.最后,利用上近似数和关系近似数相等的方式构造出了一种拟阵结构,并讨论了这一拟阵结构的一些基本性.

Binary relation, an important structure in mathematics, has been the basis of some fields. Matroid theory borrows extensively from linear algebra and graph theory,which has been widely used in many fields with abundant theories and a perfect system. By imitating the upper approximation number,a relational approximation number is defined via binary relation in this paper. Then the relational approximation number is proved to satisfy submodular for all subsets of a universe. Meanwhile, a method is proposed for calculating relational approximation number of a set by counting the cards of predecessor neighborhoods of the dements in the set. In addi- tion, by introducing a concept of a family of multisets, we give a sufficient and necessary condition under which the upper approxima- tion number of a set is equal to its relational approximation number. Finally,this paper constructs a matroidal structure in terms of the equality and discusses some fundamental properties of the structure.