Fixed-point property of posets

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.