Knaster-Tarski theorem

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.

For example, in mathematical logic least fixed points of functions on sets of formulas are used to compute the semantics of a logic program. Sometimes 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.

References

• Alfred Tarski: A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics, vol. 5 (1955), pp 285-309.

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