Kleene Fixed-point Theorem

In the mathematical areas of order and lattice theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following:

Let L be a complete partial order, and let f : L → L be a Scott-continuous (and therefore monotone) function. Then f has a least fixed point, which is the supremum of the ascending Kleene chain of f.

The ascending Kleene chain of f is the chain

obtained by iterating f on the least element ⊥ of L. Expressed in a formula, the theorem states that

where denotes the least fixed point.

This result is often attributed to Alfred Tarski, but Tarski's fixed point theorem pertains to monotone functions on complete lattices.