# 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 $S$ if and only if 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.

$\blacksquare$

## Also see

## Source of Name

This entry was named for Bronisław Knaster and Alfred Tarski.

## Historical Note

Despite the fact that the **Knaster-Tarski Theorem** bears the name of both Bronisław Knaster and Alfred Tarski, it appears that Tarski claims sole credit.

## Sources

- 1955: Alfred Tarski:
*A lattice-theoretical fixpoint theorem and its applications*(*Pacific J. Math.***Vol. 5**,*no. 2*: pp. 285 – 309): Theorem $1$