Knaster-Tarski Lemma/Corollary

Theorem

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

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

Then $f$ has a fixed point

Power Set

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $f: \powerset S \to \powerset S$ be a $\subseteq$-increasing mapping.

That is, suppose that for all $T, U \in \powerset S$:

$T \subseteq U \implies \map f T \subseteq \map f U$

Then $f$ has a fixed point.

Proof

By the Knaster-Tarski Lemma, $f$ has a least fixed point.

Thus it has a fixed point.

$\blacksquare$