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

- Publisher: Cambridge University Press
- DOI: https://doi.org/10.1017/CBO9780511810282.005
- pp 52-69

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) Let 〈X, ≤〉 be a complete lattice and f an order-preserving map 〈X, ≤〉 → 〈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 x ∈ A, we have f(x) ≤ x, whence f2(x) ≤ f(x), so f(x) ∈ A and a ≤ f(x). But f(x) ≤ x so f(a) ≤ x as desired. But a was the greatest lower bound, so f(a) ≤ a and a ∈ A. But then f(a) ∈ a since f“A ⊆ A, and f(a) ≥ a since a is the greatest lower bound.

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