Knaster-Tarski Theorem

Theorem
Let $\struct {L, \preceq}$ be a complete lattice.

Let $f: L \to L$ be an increasing mapping.

Let $F$ be the set (or class) of fixed points of $f$.

Then $\struct {F, \preceq}$ is a complete lattice.

Proof
Let $S \subseteq F$.

Let $s = \bigvee S$ be the supremum of $S$.

We wish to show that there is an element of $F$ that succeeds all elements of $S$ and is the smallest element of $F$ to do so.

By the definition of supremum, an element succeeds all elements of $S$ it succeeds $s$.

Let $U = s^\succeq$ be the upper closure of $s$.

Thus we seek the smallest fixed point of $f$ that lies in $U$.

Note that $U = \closedint s \top$, the closed interval between $s$ and the top element of $L$.

First we show that $U$ is closed under $f$.

We have that:
 * $\forall a \in S: a \preceq s$

so:
 * $a = \map f a \preceq \map f s$

Thus $\map f s$ is an upper bound of $S$.

So by the definition of supremum, $s \preceq \map f s$.

Let $x \in U$.

Then $s \preceq x$.

So:
 * $\map f s \preceq \map f x$

Since $s \preceq \map f s$, it follows that:
 * $s \preceq \map f x$

so:
 * $\map f x \in U$

Thus the restriction of $f$ to $U$ is an increasing mapping from $U$ to $U$.

By Closed Interval in Complete Lattice is Complete Lattice, $\struct {U, \preceq}$ is a complete lattice.

Thus by Knaster-Tarski Lemma, $f$ has a smallest fixed point in $U$.

Thus $S$ has a supremum in $F$.

A similar argument shows that $S$ has an infimum in $F$.

Since this holds for all $S \subseteq F$, it follows that $\struct {F, \preceq}$ is a complete lattice.

Also see

 * Knaster-Tarski Lemma
 * Knaster-Tarski Lemma: Power Set form