In combinatory logic for computer science, a fixed-point combinator (or fixpoint combinator) 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.,

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

In the classical untyped lambda calculus, every function has a fixed point. A particular implementation of ${\displaystyle \mathrm {fix} }$ is Haskell Curry's paradoxical combinator Y, given by

(Here using the standard notations and conventions of lambda calculus: Y is a function that takes one argument f and returns the entire expression following the first period; the expression ${\displaystyle \lambda x.f\ (x\ x)}$ denotes a function that takes one argument x, thought of as a function, and returns the expression ${\displaystyle f\ (x\ x)}$, where ${\displaystyle (x\ x)}$ denotes x applied to itself. Juxtaposition of expressions denotes function application, is left-associative, and has higher precedence than the period.)

The following calculation verifies that ${\displaystyle Yg}$ is indeed a fixed point of the function ${\displaystyle g}$:

The lambda term ${\displaystyle g\ (Y\ g)}$ may not, in general, β-reduce to the term ${\displaystyle Y\ g}$. However, both terms β-reduce to the same term, as shown.

Applied to a function with one variable, the Y combinator usually does not terminate. More interesting results are obtained by applying the Y combinator to functions of two or more variables. The added variables may be used as a counter, or index. The resulting function behaves like a while or a for loop in an imperative language.