By a directed set in a poset $(P, \leqslant)$, we mean a nonempty subset $D$ such that any pair $\{x, y\}$ of elements of $D$ has an upper bound in $D$. We say $(P, \leqslant)$ is directed-complete if each directed subset $D \subseteq P$ has a least upper bound in $P$. Show that a poset is complete if and only if it is directed-complete and has joins for all its finite subsets. Show also that, for any two sets $A$ and $B$, the set $[A>B]$ of partial functions from $A$ to $B$, ordered by extension, is directed-complete.

Let $(P, \leqslant)$ be a directed-complete poset, and $f: P \rightarrow P$ an order-preserving map which is inflationary, i.e. satisfies $x \leqslant f(x)$ for all $x \in P$. We define a subset $C \subseteq P$ to be closed if it satisfies $(x \in C) \rightarrow(f(x) \in C)$, and is also closed under joins of directed sets (i.e., $D \subseteq C$ and $D$ directed imply $\bigvee D \in C$ ). We write $x \ll y$ to mean that every closed set containing $x$ also contains $y$. Show that $\ll$ is a partial order on $P$, and that $x \ll y$ implies $x \leqslant y$. Now consider the set $H$ of all functions $h: P \rightarrow P$ which are order-preserving and satisfy $x \ll h(x)$ for all $x$. Show that $H$ is closed under composition of functions, and deduce that, for each $x \in P$, the set $H_{x}=\{h(x) \mid h \in H\}$ is directed. Defining $h_{0}(x)=V H_{x}$ for each $x$, show that the function $h_{0}$ belongs to $H$, and deduce that $h_{0}(x)$ is the least fixed point of $f$ lying above $x$, for each $x \in P$.