Knaster–Tarski theorem

In the mathematical areas of order and lattice theory, the Knaster–Tarski theorem, named after Bronisław Knaster and Alfred Tarski, states the following:

Let (L, ≤) be a complete lattice and let f : L → L be an order-preserving (monotonic) function w.r.t. ≤ . Then the set of fixed points of f in L forms a complete lattice under ≤ .

It was Tarski who stated the result in its most general form, and so the theorem is often known as Tarski's fixed-point theorem. Some time earlier, Knaster and Tarski established the result for the special case where L is the lattice of subsets of a set, the power set lattice.

The theorem has important applications in formal semantics of programming languages and abstract interpretation, as well as in game theory.

A kind of converse of this theorem was proved by Anne C. Davis: If every order-preserving function f : L → L on a lattice L has a fixed point, then L is a complete lattice.