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 function. Then the set of fixed points of f in L is also a complete lattice.

It was Tarski who stated the result in its most general form,^[1] 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.^[2]

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.^[3]

Consequences: least and greatest fixed points

Since complete lattices cannot be empty (they must contain supremum of empty set), the theorem in particular guarantees the existence of at least one fixed point of f, and even the existence of a least (or greatest) 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) = x, or, equivalently, such that f(x) ≤ x; the dual holds for the greatest fixpoint, the greatest element x such that f(x) = x.

If f(lim x_n)=lim f(x_n) for all ascending sequences x_n, then the least fixpoint of f is lim fⁿ(0) where 0 is the least element of L, thus giving a more "constructive" version of the theorem. (See: Kleene fixed-point theorem.) More generally, if f is monotonic, then the least fixpoint of f is the stationary limit of f^α(0), taking α over the ordinals, where f^α is defined by transfinite induction: f^α+1 = f ( f^α) and f^γ for a limit ordinal γ is the least upper bound of the f^β 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 Cantor–Bernstein–Schroeder theorem.^[4]

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:

Let L be a partially ordered set with the smallest element (bottom) and let f : L → L be an order-preserving function. Further, suppose there exists u in L such that f(u) ≤ u and that any chain in the subset {x in L : x ≤ f(x), x ≤ u} has supremum. Then f admits the least fixed point.

This can be applied to obtain various theorems on invariant sets, e.g. the Ok's theorem:

For the monotone map F : P(X) → P(X) on the family of (closed) nonempty subsets of X the following are equivalent: (o) F admits A in P(X) s.t. $A \subseteq F(A)$ , (i) F admits invariant set A in P(X) i.e. $A = F(A)$ , (ii) F admits maximal invariant set A, (iii) F admits the greatest invariant set A.

In particular, using the Knaster-Tarski principle one can develop the theory of global attractors for noncontractive discontinuous (multivalued) 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 $\langle L, \le \rangle$ and a monotone function $f\colon L \rightarrow L$ on L, the set of all fixpoints of f is also a complete lattice $\langle P, \le \rangle$ , with:

$\bigvee P = \bigvee \{ x \in L \mid x \le f(x) \}$ as the greatest fixpoint of f
$\bigwedge P = \bigwedge \{ x \in L \mid x \ge f(x) \}$ as the least fixpoint of f.

Proof. We begin by showing that P has least and greatest element. Let D = { x | x ≤ f(x) } and x ∈ D (we know that at least 0_L belongs to D). Then because f is monotone we have f(x) ≤ f(f(x)), that is f(x) ∈ D.

Now let $u = \bigvee D$ (u exists because D ⊆ L). Then for all x ∈ D it is true that x ≤ u and f(x) ≤ f(u), so x ≤ f(x) ≤ f(u). Therefore, f(u) is an upper bound of D, but u is the least upper bound, so u ≤ f(u), i.e. u ∈ D. Then f(u) ∈ D (because f(u) ≤ f(f(u))) and so f(u) ≤ u from which follows f(u) = 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 (complete) lattice $\langle L^{op}, \ge \rangle$ . As we have just proved, its greatest fixpoint there exists. It is the least one on L, so P has least and greatest elements, or more generally that every monotone function on a complete lattice has least and greatest fixpoints.

If a ∈ L and b ∈ L, we'll write [a, b] for the closed interval with bounds a and b: { x ∈ L | a ≤ x ≤ b }. If a ≤ b, then [a, b] is a complete lattice.

It remains to be proven that P is a complete lattice. Let $1_L = \bigvee L$ , W ⊆ P and $w = \bigvee W$ . We′ll show that f([w, 1_L]) ⊆ [w, 1_L]. Indeed, for every x ∈ W we have x = f(x) ≤ f(w). Since w is the least upper bound of W, w ≤ f(w). Then from y ∈ [w, 1_L] follows that w ≤ f(w) ≤ f(y), giving f(y) ∈ [w, 1_L] or simply f([w, 1_L]) ⊆ [w, 1_L]. This allows us to look at f as a function on the complete lattice [w, 1_L]. 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, which turns P into a complete lattice.

Notes

↑ Alfred Tarski (1955). "A lattice-theoretical fixpoint theorem and its applications". Pacific Journal of Mathematics. 5:2: 285–309.
↑ B. Knaster (1928). "Un théorème sur les fonctions d'ensembles". Ann. Soc. Polon. Math. 6: 133–134. With A. Tarski.
↑ Anne C. Davis (1955). "A characterization of complete lattices". Pacific J. Math. 5: 311–319. doi:10.2140/pjm.1955.5.311.
↑ Example 3 in R. Uhl, "Tarski's Fixed Point Theorem", from MathWorld--a Wolfram Web Resource, created by Eric W. Weisstein.

References

Andrzej Granas and James Dugundji (2003). Fixed Point Theory. Springer-Verlag, New York. ISBN 0-387-00173-5.
Forster, T. Logic, Induction and Sets. ISBN 0-521-53361-9.

External links

J. B. Nation, Notes on lattice theory.

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