A Lattice-Theoretical Fixpoint Theorem and Its Applications.

Tarski, A. (1955). A Lattice-Theoretical Fixpoint Theorem and Its Applications. Pacific Journal of Mathematics, 5(2), 285-309.

Type

Journal article

Intellectual base

Primary research

Year

1955

Volume

5

Issue

2

Pages

285-309

DOI

10.2140/pjm.1955.5.285

DOI registrant

Mathematical Sciences Publishers

Link

https://doi.org/10.2140/pjm.1955.5.285

Cited from

mathematics

Cited by

3 citations across 3 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Fixed Point
  • When an object is defined in terms of itself — the prediction that shapes the thing predicted, the value of a recursive definition, the price that depends on the price — the object's existence is the existence of a fixed point of an associated map, and many apparent paradoxes dissolve once the underlying self-referential map and its fixed points are made explicit.
• Order
  • or fixed-point analysis (the Knaster-Tarski theorem on monotone maps over complete lattices, which Tarski (1955) generalized into the form now used in denotational semantics and abstract interpretation)

    This sourceSource of the Knaster-Tarski fixed-point theorem as now formulated. Precursor: Knaster, Bronisław, and Alfred Tarski. "Un théorème sur les fonctions d'ensembles." Annales de la Société Polonaise de Mathématique 6 (1928): 133–134.

    Supported in partVerified against the work's full text

    Tarski confirms that his lattice theorem generalizes the earlier Knaster-Tarski set-theoretic result for increasing functions; it says nothing of denotational semantics or abstract interpretation.

    “In 1927 Knaster and the author proved a set-theoretical fixpoint theorem ... A generalization of this result is the lattice-theoretical fixpoint theorem stated above as Theorem 1.”

    Read by an automated reader; not a human review. How support was checked

• Relation
  • Similarly, a partial order is characterized by reflexivity, antisymmetry, and transitivity; these properties together license the Hasse-diagram visualization, the least-upper-bound construction, and the order-theoretic fixed-point theorems (Knaster-Tarski

    This sourceSource of the Knaster-Tarski fixed-point theorem as now formulated. Precursor: Knaster, Bronisław, and Alfred Tarski. "Un théorème sur les fonctions d'ensembles." Annales de la Société Polonaise de Mathématique 6 (1928): 133–134.

Verification

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