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dbr:Knaster–Tarski_theorem

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Knaster–Tarski theorem

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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 monotonic function (w.r.t. ≤). Then the set of fixed points of f in L also 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.

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dbo:abstract

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 monotonic function (w.r.t. ≤). Then the set of fixed points of f in L also 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. 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.

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n12:Knaster–Tarski_theorem