## least fixed point

A function f may have many fixed points (x such that f x = x). For example, any value is a fixed point of the identity function, (\ x . x).

If f is recursive, we can represent it as

f = fix F

where F is some higher-order function and

fix F = F (fix F).

The standard denotational semantics of f is then given by the least fixed point of F. This is the least upper bound of the infinite sequence (the ascending Kleene chain) obtained by repeatedly applying F to the totally undefined value, bottom. I.e.

fix F = LUB {bottom, F bottom, F (F bottom), ...}.

The least fixed point is guaranteed to exist for a continuous function over a cpo.