Fixed points of continuous DCPOs (Q444683)

The authors first classify continuous (in the Scott topology) selfmaps of dcpos (directed complete partial orders) by finding equivalences when the fixed points of continuous selfmaps themselves form a dcpo. This also characterizes dcpos and continuous dcpos. In particular the authors show that if \(X\) is a continuous dcpo then the set of fixed points of the selfmap \(f\), Fix(\(f\)), is a dcpo but not necessarily continuous. Next, a concrete example when Fix(\(f\)) is not continuous is given. The authors proceed to characterize the continuous dcpos \(X\) for which Fix(\(f\)) is continuous for every continuous \(f\). In the next section, several results showing when Fix(\(f\)) is a continuous dcpo are given in terms of its structure. The paper concludes with a look at locally stable maps and the result that if \(D\) is a continuous poset and \(f:D\rightarrow D\) is locally stable, then Fix(\(f\)) is a continuous poset.