论文标题
在强大集的等级函数上
On the Rank Functions of Powerful Sets
论文作者
论文摘要
如果对于所有$ x \ subseteq e $，则有限集$ e $的子集的$ s \ subseteq 2^e $是\ emph {功能}，$ s $中的$ x $的子集的数量是2个功率。每个功能均为2。每个强大的集合都与非负性integer integer integer integer valued函数相关联，我们呼吁我们呼吁等级函数。 Farr和Wang引入了强大的集合，作为二进制矩阵的概括，因为二进制Matroid的Cocircuit空间具有相应的Matroid Rank函数的强大集合。 在本文中，我们研究了如何通过其等级函数来表征强大集合的结构属性。强大的集合具有四种类型的退化元素，包括循环和配色。我们表明，强大集合的等级函数的某些评估确定了退化元素。我们介绍了强大的多集，并证明了它们的一些基本结果。我们表明，当且仅当其等级函数是亚心下方时，强大的集合就对应于二进制曲线。本文回答了Farr和Wang在肯定中做出的两个猜想。
A set $S\subseteq 2^E$ of subsets of a finite set $E$ is \emph{powerful} if, for all $X\subseteq E$, the number of subsets of $X$ in $S$ is a power of 2. Each powerful set is associated with a non-negative integer valued function, which we call the rank function. Powerful sets were introduced by Farr and Wang as a generalisation of binary matroids, as the cocircuit space of a binary matroid gives a powerful set with the corresponding matroid rank function. In this paper we investigate how structural properties of a powerful set can be characterised in terms of its rank function. Powerful sets have four types of degenerate elements, including loops and coloops. We show that certain evaluations of the rank function of a powerful set determine the degenerate elements. We introduce powerful multisets and prove some fundamental results on them. We show that a powerful set corresponds to a binary matroid if and only if its rank function is subcardinal. This paper answers the two conjectures made by Farr and Wang in the affirmative.
