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The Lefschetz fixed-point theorem and the Nielsen fixed-point theorem from algebraic topology give a way to count fixed points. In Galois theory , the set of the fixed points of a set of field automorphisms is a field called the fixed field of the set of automorphisms.
The FPP is a topological invariant , i. is preserved by any homeomorphism. The FPP is also preserved by any retraction.
According to the Brouwer fixed-point theorem , every compact and convex subset of a Euclidean space has the FPP. Compactness alone does not imply the FPP, and convexity is not even a topological property, so it makes sense to ask how to topologically characterize the FPP.
In Borsuk asked whether compactness together with contractibility could be a necessary and sufficient condition for the FPP to hold.
The problem was open for 20 years until the conjecture was disproved by Kinoshita who found an example of a compact contractible space without the FPP.
In domain theory , the notion and terminology of fixed points is generalized to a partial order. Prefixpoints and postfixpoints have applications in theoretical computer science.
In order theory , the least fixed point of a function from a partially ordered set poset to itself is the fixed point which is less than each other fixed point, according to the order of the poset.
A function need not have a least fixed point, but if it does then the least fixed point is unique. One way to express the Knaster—Tarski theorem is to say that a monotone function on a complete lattice has a least fixpoint that coincides with its least prefixpoint and similarly its greatest fixpoint coincides with its greatest postfixpoint.
Formally, if the function f has one or more fixed points, then. In mathematical logic , fixed-point logics are extensions of classical predicate logic that have been introduced to express recursion. Their development has been motivated by descriptive complexity theory and their relationship to database query languages , in particular to Datalog.
In many fields, equilibria or stability are fundamental concepts that can be described in terms of fixed points. Some examples follow. Contents move to sidebar hide. Article Talk. Read Edit View history. Tools Tools. What links here Related changes Upload file Special pages Permanent link Page information Cite this page Get shortened URL Download QR code Wikidata item.
Download as PDF Printable version. In other projects. Wikimedia Commons. Element mapped to itself by a mathematical function. Main article: Fixed-point iteration. Main article: Fixed-point theorems.
Main article: Fixed-point property. Main article: Least fixed point. Main article: Fixed point combinator.
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