Note about Fixed Points of Scott Continuous Self-Mappings

Abstract:

      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 $L$.