# Knaster-Tarski Lemma/Power Set

## Theorem

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 greatest fixed point and a least fixed point.

## Proof

By Power Set is Complete Lattice, $\struct {\powerset S, \cap, \cup, \subseteq}$ is a complete lattice.

Thus the theorem holds by the Knaster-Tarski Lemma.

$\blacksquare$

## Source of Name

This entry was named for Bronisław Knaster and Alfred Tarski.