Fixed-point property of posets

Summary

A partially ordered set has the fixed-point property if every order-preserving map from it to itself fixes some element. It is the order-theoretic counterpart of the fixed-point property of topological spaces. Every complete lattice has it by the Knaster–Tarski theorem, and among lattices exactly the complete ones have it. Deciding which finite posets have the fixed-point property is a hard combinatorial question.