Scott连续自映射不动点的注记

Note about Fixed Points of Scott Continuous Self-Mappings

中文关键词:  Scott连续  不动点  稳定映射.

英文关键词:Scott continuous  fixed points  stable mapping.

中文摘要:

      本文讨论了连续domain $L$ 在什么条件下,存在Scott连续自映射$f:L\rightarrow L$ 使得不动点集 ${\rm fix}(f)$在$L$的诱导序下不连续,对有可数基且有最小元的代数domain $L$,部分回答了Huth提出的一个问题.同时,给出了一个例子表明,存在有界完备 domain $L$,使得对 $L$ 间的任何Scott连续稳定自映射 $f$, ${\rm fix}(f)$ 不是 $L$ 的收缩.

英文摘要:

      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 $L$.