# Fixed-point combinator

## Higher-order function Y for which Y f = f (Y f) / From Wikipedia, the free encyclopedia

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In combinatory logic for computer science, a **fixed-point combinator** (or **fixpoint combinator**),[1]^{: p.26 } is a higher-order function (i.e. a function which takes a function as argument) that returns some *fixed point* (a value that is mapped to itself) of its argument function, if one exists.

Formally, if ${\textrm {fix}}$ is a fixed-point combinator and the function $f$ has one or more fixed points, then ${\textrm {fix}}\ f$ is one of these fixed points, i.e.

- $f\ ({\textrm {fix}}\ f)={\textrm {fix}}\ f\ .$

Fixed-point combinators can be defined in the lambda calculus and in functional programming languages and provide a means to allow for recursive definitions.

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