Fixed point combiner

A fixed point combiner is a mathematical operator in the form of a higher-order function , which returns one of its fixed points from a function . A fixed point of the function fulfills the condition ${\displaystyle {\hat {y}}}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle f}$

${\displaystyle x=f(x)}$

and thus

${\displaystyle {\hat {y}}f=x.}$

By inserting the property of the fixed point combiner then results as

${\displaystyle {\hat {y}}\ f=f\ ({\hat {y}}\ f).}$

Special fixed point combiners

A special fixed point combiner is the Y-combiner of the lambda calculus described by Haskell Curry :

${\displaystyle \lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))}$

Fixed point combiners are used in various areas:

• in the untyped and in the typed lambda calculus ,
• in functional programming
• and in imperative programming .