Tarski Fixed Point Theorem For Compact Euclidean Lattices

Let $L$ be a compact nonempty lattice in $\mathbb R^n$ with the coordinatewise partial order. If $$ f:L\to L $$ is monotone, then $f$ has a fixed point.

Moreover, every nonempty subset of $L$ has a supremum and an infimum in $L$, so $L$ has a greatest and a smallest element.

References

• [MFoGT, Section 4.12, Exercise 4(1)] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. Tarski fixed point theorem for compact nonempty lattices in Euclidean space.