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:
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.
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.

Consequences: least and greatest fixed points

Since complete lattices cannot be empty, the theorem in particular guarantees the existence of at least one fixed point of f, and even the existence of a least fixed point. In many practical cases, this is the most important implication of the theorem.
The least fixpoint of f is the least element x such that f = x, or, equivalently, such that f ≤ x; the dual holds for the greatest fixpoint, the greatest element x such that f = x.
If f=lim f for all ascending sequences xn, then the least fixpoint of f is lim fn where 0 is the least element of L, thus giving a more "constructive" version of the theorem. More generally, if f is monotonic, then the least fixpoint of f is the stationary limit of fα, taking α over the ordinals, where fα is defined by transfinite induction: fα+1 = f and fγ for a limit ordinal γ is the least upper bound of the fβ for all β ordinals less than γ. The dual theorem holds for the greatest fixpoint.
For example, in theoretical computer science, least fixed points of monotone functions are used to define program semantics. Often a more specialized version of the theorem is used, where L is assumed to be the lattice of all subsets of a certain set ordered by subset inclusion. This reflects the fact that in many applications only such lattices are considered. One then usually is looking for the smallest set that has the property of being a fixed point of the function f. Abstract interpretation makes ample use of the Knaster–Tarski theorem and the formulas giving the least and greatest fixpoints.
Knaster–Tarski theorem can be used for a simple proof of the Cantor–Bernstein–Schroeder theorem.

Weaker versions of the theorem

Weaker versions of the Knaster–Tarski theorem can be formulated for ordered sets, but involve more complicated assumptions. For example:
This can be applied to obtain various theorems on invariant sets, e.g. the Ok's theorem:
In particular, using the Knaster-Tarski principle one can develop the theory of global attractors for noncontractive discontinuous iterated function systems. For weakly contractive iterated function systems Kantorovitch fixpoint theorem suffices.
Other applications of fixed point principles for ordered sets come from the theory of differential, integral and operator equations.

Proof

Let's restate the theorem.
For a complete lattice and a monotone function on L, the set of all fixpoints of f is also a complete lattice, with:
Proof. We begin by showing that P has both a least element and a greatest element. Let D = and x ∈ D. Then because f is monotone we have f ≤ f, that is f ∈ D.
Now let . Then for all x ∈ D it is true that x ≤ u and f ≤ f, so x ≤ f ≤ f. Therefore, f is an upper bound of D, but u is the least upper bound, so u ≤ f, i.e. u ∈ D. Then f ∈ D and so f ≤ u from which follows f = u. Because every fixpoint is in D we have that u is the greatest fixpoint of f.
The function f is monotone on the dual lattice. As we have just proved, its greatest fixpoint exists. It is the least fixpoint of L, so P has least and greatest elements, that is more generally, every monotone function on a complete lattice has a least fixpoint and a greatest fixpoint.
If a ∈ L and b ∈ L, we'll write for the closed interval with bounds a and b:. If a ≤ b, then , is a complete lattice.
It remains to be proven that P is a complete lattice. Let, W ⊆ P and. We′ll show that f ⊆ . Indeed, for every x ∈ W we have x = f and since w is the least upper bound of W x ≤ f. In particular w ≤ f. Then from y ∈ follows that w ≤ f ≤ f, giving f ∈ or simply f ⊆ . This allows us to look at f as a function on the complete lattice . Then it has a least fixpoint there, giving us the least upper bound of W. We′ve shown that an arbitrary subset of P has a supremum, that is, P is a complete lattice.