| 张华军.套链分解[J].数学研究及应用,2008,28(1):39~45 |
| 套链分解 |
| Nested Chain Order |
| 投稿时间：2005-12-26 修订日期：2006-03-02 |
| DOI：10.3770/j.issn:1000-341X.2008.01.006 |
| 中文关键词: |
| 英文关键词:poset normalized matching property sperner property nested chain decomposition. |
| 基金项目:国家自然科学基金(No. 10471016). |
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| 中文摘要: |
| 假定$S$是一个集合, $X_1,X_2,\ldots,X_k$是$S$的$k$个互不相交且秩为$m$的子集.定义$H(m,k)=\{X\subseteq S: X$不包含在任何一个$X_i$里面$\}$和%$P(m,k)=\{X\subseteq S: X$至少与两个$X_i$相交非空$\}$.当$S=\bigcup_{1\leq i\leq k}X_i$, 我们定义$Q(m,k,2)=\{X\subseteq S$: 至少存在一个$X_i$使得$|X\cap X_i|\geq 2\}$. 设$Y_1$和$Y_2$是$S$的两个互不相交的子集,定义${\cal F}_{1,j}=\{X\subseteq S: |X\cap Y_1|\geq 1$ 或者$|X\cap Y_2|\geq j\}$.显然在包含关系下这四个偏序集都是分次偏序集.在本文我们将证明这四个偏序集都有套链分解. |
| 英文摘要: |
| Let $X_{1},X_{2},\ldots,X_{k}$ be $k$ disjoint subsets of $S$ with the same cardinality $m$. Define $H(m,k)=\{X\subseteq S: X \not\subseteq X_i$ for $1\leq i\leq k\}$ and $P(m,k)=\{X\subseteq S: X \cap X_i \neq \emptyset$ for at least two $X_i$'s\}. Suppose $S=\bigcup_{i=1}^k X_i$, and let $Q(m,k,2)$ be the collection of all subsets $K$ of $S$ satisfying $|K\cap X_{i}|\geq 2$ for some $1 \leq i \leq k$. For any two disjoint subsets $Y_1$ and $Y_2$ of $S$, we define ${\cal F}_{1,j}=\{X\subseteq S: \mbox{either $|X\cap Y_1|\geq 1$ or $|X\cap Y_2|\geq j$}\}$. It is obvious that the four posets are graded posets ordered by inclusion. In this paper we will prove that the four posets are nested chain orders. |
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