---
tag: QTQN93B305
kind: definition
name: Fixed-point property of posets
msc: [06A06, 47H10]
defines: HMDGNS1V2T
uses: [0QV2JS9RJ3, 0A7KV2WB7A]
sources:
  - url: https://en.wikipedia.org/wiki/Fixed-point_property
---

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
$P$ has the *fixed-point property* if every [[monotone|0A7KV2WB7A]] map $f \colon P \to P$ has a fixed point, that is, an element $x \in P$ with $f(x) = x$.

## Remarks

Example: every finite chain $c_1 < c_2 < \cdots < c_n$ has the fixed-point property.
For monotone $f$, $c_1 \le f(c_1)$, and induction gives $f^k(c_1) \le f^{k+1}(c_1)$; this increasing sequence in a finite set is eventually constant, at a fixed point.
Non-example: the two-element antichain $\set{a, b}$ fails it, since the map exchanging $a$ and $b$ is monotone and has no fixed point.
The chain $\Z$ also fails it, with the monotone map $x \mapsto x + 1$.
Every complete lattice has the fixed-point property, by the Knaster–Tarski theorem.
More generally, every finite poset with a least element has it, by the same iteration as for chains.
The property is preserved by order isomorphisms and passes to retracts.