| A Note of Order Congruences on Ordered Semigroups |
| Received:August 31, 2006 Revised:March 23, 2007 |
| Key Words:
ordered semigroup order congruence convex ideal.
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| Fund Project:the National Natural Science Foundation of China (Nos.10626012; 103410020); the Jiangsu Planned for Postdoctoral Research Found (No.0502022B); the Natural Science Foundation of Guangdong Province (No.0501332); the Educational Department Natural Science Fo |
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| Abstract: |
| Which subset of an ordered semigroup $S$ can serve as a congruence class of certain order-congruence on $S$ is an important problem. XIE Xiangyun proved that if every ideal $C$ of an ordered semigroup $S$ is a congruence class of one order-congruence on $S$, then $C$ is convex and when $C$ is strongly convex, the reverse statement is true in 2001. In this paper, we give an alternative constructing order congruence method, and we prove that every ideal $B$ is a congruence class of one order congruence on $S$ if and only if $B$ is convex. Furthermore, we show that the order relation defined by this method is ``the least'' order congruence containing $B$ as a congruence class. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2008.04.020 |
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