Fixed-point combinator

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

${\displaystyle \mathrm {fix} \ f\ =f\ (\mathrm {fix} \ f).}$

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