Note about Fixed Points of Scott Continuous Self-Mappings
Received:August 10, 2008  Revised:September 24, 2008
Key Word: Scott continuous   fixed points   stable mapping.
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.10571112) and the National Key Project of Fundamental Research (Grant No.2002CB312200).
 Author Name Affiliation Xiao Yong XI College of Mathematics and Information Science, Shaanxi Normal University, Shaanxi 710062, P. R. China College of Mathematics Science, Xuzhou Normal University, Jiangsu 221009, P. R. China Yong Ming LI College of Computer Science, Shaanxi Normal University, Shaanxi 710062, P. R. China
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It is discussed in this paper that under what conditions, for a continuous domain $L$, there is a Scott continuous self-mapping $f:L\rightarrow L$ such that the set of fixed points ${\rm fix}(f)$ is not continuous in the ordering induced by $L$. For any algebraic domain $L$ with a countable base and a smallest element, the problem presented by Huth is partially solved. Also, an example is given and shows that there is a bounded complete domain $L$ such that for any Scott continuous stable self-mapping $f$, ${\rm fix}(f)$ is not the retract of $L$.