A Note of Order Congruences on Ordered Semigroups
Received:August 31, 2006  Revised:March 23, 2007
Key Words: ordered semigroup   order congruence   convex ideal.  
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
Author NameAffiliation
SHI Xiao Ping Department of Mathematics, Nanjing Normal University, Jiangsu 210046, China
Department of Mathematics, Southeast University, Jiangsu 210096, China 
XIE Xiang Yun Department of Mathematics and Physics, Wuyi University, Guangdong 529020, China 
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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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