| On the Set of Common Consequent Indices of a Class of Binary Relations |
| Received:July 19, 2006 Revised:December 12, 2006 |
| Key Words:
common consequent index primitive relation directed graph.
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| Fund Project:the Natural Science Foundation of Jiangsu Province (No.BK2007030); the Natural Science Foundation of Education Committee of Jiangsu Province (No.07KJD110207). |
| Author Name | Affiliation | | MA Hong Ping | School of Mathematical Sciences, Xuzhou Normal University, Jiangsu 221116, China | | MIAO Zheng Ke | School of Mathematical Sciences, Xuzhou Normal University, Jiangsu 221116, China
Institute of Applied Mathematics, Academy of Mathematics and System Science, Chinese Academy of Science, Beijing 100080, China |
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| Abstract: |
| 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. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2008.03.002 |
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