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  • Print publication year: 2003
  • Online publication date: June 2012

3 - Partially ordered sets

Summary

Lattice fixed point theorems

A fixed point for a function f is an argument x such that f(x) = x. This is an important concept because many useful mathematical facts can be expressed by assertions that say that certain functions have fixed points. For example, the equation p(x) = 0 has a solution iff the function λx.(p(x) – x) has a fixed point. This gives us a motive to seek methods for showing that functions have fixed points: fixed point theorems are useful in the search for solutions to equations.

The Tarski-Knaster theorem

THEOREM 8 (The Tarski-Knaster theorem) LetX, ≤〉 be a complete lattice and f an order-preserving mapX, ≤〉 → 〈X, ≤〉. Then f has a fixed point.

Proof: Set A = {x : f(x)x} and a = ∧A. (A is nonempty because it must contain ∨X.) Since f is order-preserving, we can say that if f(x)x, then f2(x)f(x), and so f(a) is also a lower bound for A as follows. If xA, we have f(x)x, whence f2(x)f(x), so f(x)A and af(x). But f(x)x so f(a)x as desired. But a was the greatest lower bound, so f(a)a and aA. But then f(a)a since fAA, and f(a)a since a is the greatest lower bound.