Explain what is meant by a chain-complete poset. State the Bourbaki-Witt fixedpoint theorem for such posets.

A poset $P$ is called directed if every finite subset of $P$ (including the empty subset) has an upper bound in $P ; P$ is called directed-complete if every subset of $P$ which is directed (in the induced ordering) has a least upper bound in $P$. Show that the set of all chains in an arbitrary poset $P$, ordered by inclusion, is directed-complete.

Given a poset $P$, let $[P \rightarrow P]$ denote the set of all order-preserving maps $P \rightarrow P$, ordered pointwise (i.e. $f \leqslant g$ if and only if $f(x) \leqslant g(x)$ for all $x$ ). Show that $[P \rightarrow P]$ is directed-complete if $P$ is.

Now suppose $P$ is directed-complete, and that $f: P \rightarrow P$ is order-preserving and inflationary. Show that there is a unique smallest set $C \subseteq[P \rightarrow P]$ satisfying

(a) $f \in C$;

(b) $C$ is closed under composition (i.e. $g, h \in C \Rightarrow g \circ h \in C$ ); and

(c) $C$ is closed under joins of directed subsets.

Show that

(i) all maps in $C$ are inflationary;

(ii) $C$ is directed;

(iii) if $g=\bigvee C$, then all values of $g$ are fixed points of $f$;

(iv) for every $x \in P$, there exists $y \in P$ with $x \leqslant y=f(y)$.