Exercise. [dt-000Q]

Give an example of a poset A and a monotonic function f : A \rightarrow A such that f doesn't have a fixed point.
Solution.
- Liam O'Connor
In the poset $(\mathbb {N},\leq )$ , the function $f(x) = x + 1$ has no fixed points.
Give an example of a poset $A$ and a monotonic function $f : A \rightarrow A$ such that $f$ has multiple fixed points.

Solution.

In the poset $(\mathbb {N},\leq )$ , the function $f(x) = x$ has infinitely many fixed points.