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

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

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