About: 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.

Attributes	Values
rdfs:label	Knaster–Tarski theorem (en)
rdfs:comment	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. (en)
sameAs	Knaster–Tarski theorem
dbp:wikiPageUsesTemplate	dbt:Cite_book dbt:Cite_journal dbt:Reflist
Subject	Articles containing proofs Fixed-point theorems Fixed points (mathematics) Order theory Theorems in the foundations of mathematics
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Knaster–Tarski_theorem?oldid=1034816708&ns=0
Wikipage page ID	181417 (xsd:integer)
page length (characters) of wiki page	11613 (xsd:nonNegativeInteger)
Wikipage revision ID	1034816708 (xsd:integer)
Link from a Wikipage to another Wikipage	Abstract interpretation Alfred Tarski Anne C. Morel Monotonic function Semantics (computer science) Bronisław Knaster Articles containing proofs James Dugundji Fixed-point theorems Fixed points (mathematics) Order theory Duality (order theory) Power set Set (mathematics) Springer Science+Business Media Complete lattice Computer science Converse (logic) Order theory Ordinal number Partially ordered set Empty set Fixed-point subgroup Function (mathematics) Cantor–Bernstein theorem Theorems in the foundations of mathematics Infimum and supremum Invariant manifold Modal μ-calculus Transfinite induction Subset Total order Iterated function system Kleene fixed-point theorem Lattice (order) Least fixed point Mathematics Mathematische Nachrichten dbr:Kantorovitch_fixpoint_theorem
has 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. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Knaster–Tarski_theorem
is Wikipage redirect of	Tarski's fixed point theorem Tarski's fixed-point theorem Knaster-Tarski theorem Tarski-Knaster theorem Tarski fixed point theorem Tarski–Knaster theorem
is Link from a Wikipage to another Wikipage of	List of theorems List of things named after Alfred Tarski Market design Abstract interpretation Anne C. Morel Bisimulation Bronisław Knaster Denotational semantics List of functional programming topics List of order theory topics Complete lattice Fixed-point theorems Fixed-point subgroup Interval propagation Introduction to Lattices and Order Modal μ-calculus Kleene fixed-point theorem Knaster-Tarski theorem Least fixed point
is foaf:primaryTopic of	http://en.wikipedia.org/wiki/Knaster–Tarski_theorem

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