一类二元关系的公共后继指数集

On the Set of Common Consequent Indices of a Class of Binary Relations

中文关键词:  公共后继指数  本原关系  有向图.

英文关键词:common consequent index  primitive relation  directed graph.

中文摘要:

      设$V=\{ a_{1},a_{2},\ldots ,a_{n}\}$是$n\geq 2$的一个有限集合,$V$上所有本原的二元关系组成的集合记为$P_{n}(V)$.对任意的$Q\in P_{n}(V)$,与$Q$对应的有向图记为$G(Q)$.记$ P_{n}(V,d)=\{Q:Q\in P_{n}(V)$ 且$G(Q)$ 恰好包含 $d$ 个环\},其中$0

英文摘要:

      Let $V=\{ a_{1},a_{2},\ldots,a_{n}\} $ be a finite set with $n\geq 2$ and $P_{n}(V)$ the set of all primitive binary relations on $V$. For $Q\in P_{n}(V)$, denote by $G(Q)$ the directed graph corresponding to $Q$. For positive integer $d\leq n$, let $ P_{n}(V,d)=\{Q:Q\in P_{n}(V)$ and $G(Q)$ contains exactly $d$ loops\}. In this paper, it is proved that the set of common consequent indices of binary relations in $P_{n}(V,d)$ is $\{1,2,\ldots,n-\lceil \frac{d}{2}\rceil \}$. Furthermore, the minimal extremal binary relations are described.