Tarski’s fixed point theorem

E353628

mathematical theorem result in lattice theory result in order theory

Tarski’s fixed point theorem is a fundamental result in order theory and lattice theory that guarantees the existence of fixed points for monotone functions on complete lattices, with wide applications in logic, computer science, and economics.

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All labels observed (2)

Label	Occurrences
Knaster–Tarski theorem	2
Tarski’s fixed point theorem canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T3380931 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Tarski’s fixed point theorem
Context triple: [Alfred Tarski, knownFor, Tarski’s fixed point theorem]

A. Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
B. Glicksberg fixed-point theorem
The Glicksberg fixed-point theorem is a result in functional analysis that extends Kakutani’s fixed-point theorem to certain infinite-dimensional or compact convex subsets of locally convex topological vector spaces.
C. Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
D. Ky Fan’s lemma
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
E. Kripke fixed-point theory of truth
The Kripke fixed-point theory of truth is a semantic framework developed by Saul Kripke that uses partial truth predicates and fixed points to consistently handle self-referential sentences and semantic paradoxes like the liar paradox.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Tarski’s fixed point theorem
Target entity description: Tarski’s fixed point theorem is a fundamental result in order theory and lattice theory that guarantees the existence of fixed points for monotone functions on complete lattices, with wide applications in logic, computer science, and economics.

A. Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
B. Glicksberg fixed-point theorem
The Glicksberg fixed-point theorem is a result in functional analysis that extends Kakutani’s fixed-point theorem to certain infinite-dimensional or compact convex subsets of locally convex topological vector spaces.
C. Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
D. Ky Fan’s lemma
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
E. Kripke fixed-point theory of truth
The Kripke fixed-point theory of truth is a semantic framework developed by Saul Kripke that uses partial truth predicates and fixed points to consistently handle self-referential sentences and semantic paradoxes like the liar paradox.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in lattice theory ⓘ result in order theory ⓘ
appliesTo	isotone maps on complete lattices ⓘ monotone endofunctions on complete lattices ⓘ
assumptionOnFunction	monotone ⓘ
characterizes	set of fixed points as complete lattice ⓘ
domain	complete lattice ⓘ
field	denotational semantics ⓘ economics ⓘ fixed-point theory ⓘ game theory ⓘ lattice theory ⓘ mathematical logic ⓘ order theory ⓘ program verification ⓘ theoretical computer science ⓘ
generalizes	fixed-point results for monotone operators on power sets ⓘ
guarantees	existence of fixed points ⓘ existence of greatest fixed point ⓘ existence of least fixed point ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	fixed points can be obtained as limits of approximation sequences in complete lattices ⓘ
namedAfter	Alfred Tarski ⓘ
relatedTo	Banach fixed-point theorem ⓘ Brouwer fixed-point theorem ⓘ Kleene’s recursion theorem ⓘ surface form: Kleene fixed-point theorem Tarski’s fixed point theorem self-linksurface differs ⓘ surface form: Knaster–Tarski theorem
requires	existence of arbitrary joins ⓘ existence of arbitrary meets ⓘ
statedIn	order-theoretic terms ⓘ
typicalDomainExample	lattice of subsets ordered by inclusion ⓘ powerset lattice of a set ⓘ
usedIn	abstract interpretation ⓘ construction of coinductive definitions ⓘ construction of inductive definitions ⓘ dataflow analysis ⓘ definition of semantics of recursive programs ⓘ denotational semantics of programming languages ⓘ economic models of strategic complementarities ⓘ fixed-point logics ⓘ formal verification ⓘ game-theoretic models with monotone best responses ⓘ general equilibrium theory under monotonicity assumptions ⓘ model theory ⓘ proof theory ⓘ μ-calculus semantics ⓘ

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Alfred Tarski → knownFor → Tarski’s fixed point theorem ⓘ

Tarski’s fixed point theorem → relatedTo → Tarski’s fixed point theorem self-linksurface differs ⓘ

this entity surface form: Knaster–Tarski theorem

Bronisław Knaster → notableWork → Tarski’s fixed point theorem ⓘ