Whereas a FIXED-point of a first-order function (a function on "simple" values such as integers) is a first-order value, a fixed point of a higher-order function F is ANOTHER function f-fix such that

F(F-fix) = F-fix.

A fixed point operator is a function FIX which produces such a fixed point f-fix for any function F:

FIX(F) = F-fix.

Therefore: F( FIX(F) ) = FIX(F).

Fixed point combinators allow the definition of anonymous recursive functions. Somewhat surprisingly, they can be defined with non-recursive lambda ABSTRACTIONS.

Whereas a fixed-point of a first-order function (a function on "simple" values such as integers) is a first-order value, a fixed point of a higher-order function F is another function f-fix such that

F(F-fix) = F-fix.

A fixed point operator is a function FIX which produces such a fixed point f-fix for any function F:

FIX(F) = F-fix.

Therefore: F( FIX(F) ) = FIX(F).

Fixed point combinators allow the definition of anonymous recursive functions. Somewhat surprisingly, they can be defined with non-recursive lambda abstractions.