Some Fixed Point Theorems for Partially Ordered Sets

Abstract

A partially ordered set P has the fixed point property if every order-preserving map f : P → P has a fixed point, i.e. there exists x ∊ P such that f(x) = x. A. Tarski's classical result (see [4]), that every complete lattice has the fixed point property, is based on the following two properties of a complete lattice P:(A)For every order-preserving map f : P → P there exists x ∊ P such that x ≦ f(x).(B)Suprema of subsets of P exist; in particular, the supremum of the set {x|x ≦ f(x)} ⊂ P exists.