Knaster-Tarski theorem
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In the mathematical areas of order and lattice theory, the Knaster-Tarski theorem, named after Bronislaw 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.
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 (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 xn)=lim f(xn) for all xn an ascending sequence of elements of L, 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. More generally, if f is monotonic, then the least fixpoint of f is the stationary limit of fα(0), taking α over the ordinals, where fα is defined by transfinite induction: fα+1 = f ( fα) 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 compute 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.
References
- Alfred Tarski: A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics, vol. 5 (1955), pp 285-309.
- Andrzej Granas and James Dugundji, Fixed Point Theory (2003) Springer-Verlag, New York, ISBN 0-387-00173-5fr:Théorème de Knaster-Tarski