石小平,谢祥云.序半群序同余的一个注记[J].数学研究及应用,2008,28(4):898~904 |
序半群序同余的一个注记 |
A Note of Order Congruences on Ordered Semigroups |
投稿时间:2006-08-31 修订日期:2007-03-23 |
DOI:10.3770/j.issn:1000-341X.2008.04.020 |
中文关键词: 序半群 同余 凸理想. |
英文关键词:ordered semigroup order congruence convex ideal. |
基金项目:国家自然科学基金(No.10626012; 103410020); 江苏省博士后科研资助计划(No.0502022B); 广东省自然科学基金(No.0501332); 广东省教育厅科学基金(No.Z03070). |
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中文摘要: |
序半群S的什么子集可以作为S的同余类是一个重要的问题. 在文[8]中,作者证明了如果序半群S的 理想$C$是$S$的某个同余类, 则$C$是凸的; 而且当$C$是强凸理想时,逆命题成立. 在本文中, 我们给出了序半群同余的一个新的构造,并证明了序半群$S$的理想$B$是$S$的某个同余类的充要条件是$B$是凸的. |
英文摘要: |
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. |
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