---
tag: G20G1ECN56
kind: statement
role: theorem
name: Davis's converse of the Knaster–Tarski theorem
msc: [06B23, 47H10]
uses: [WH4SFAACAS, D7A0ZGZT5S, 0A7KV2WB7A, RHXP3HA64M]
about: [D7A0ZGZT5S]
sources:
  - url: https://en.wikipedia.org/wiki/Knaster%E2%80%93Tarski_theorem
---

## Statement

Let $L$ be a [[lattice|WH4SFAACAS]].
Then $L$ is a [[complete lattice|D7A0ZGZT5S]] if and only if every [[monotone function|0A7KV2WB7A]] $f \colon L \to L$ has a fixed point, that is, an element $x \in L$ with $f(x) = x$.

## Remarks

One direction is the [[Knaster–Tarski theorem|RHXP3HA64M]]: in a complete lattice, every monotone map has a fixed point, and even a complete lattice of fixed points.
For the converse, if $L$ is not complete, Davis constructs a monotone map from $L$ to $L$ without fixed points.
The lattice $\Z$ with its usual order is not complete, and the map $n \mapsto n + 1$ is monotone without fixed points.
The result was published by Anne C. Davis in 1955.

## Explanations

### L1 human

The Knaster–Tarski theorem says that in a complete lattice every order-preserving map has a fixed point.
Davis showed that this property characterizes complete lattices among all lattices.
So the existence of fixed points is not a lucky extra: it is exactly completeness.

### L2 human

The result gives a test for completeness that does not mention suprema at all.
To show that a lattice is not complete, it is enough to exhibit one monotone map without a fixed point.
A shift $n \mapsto n + 1$ on $\Z$ is such a map.