In the mathematical areas of order and lattice theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following: Kleene Fixed-Point Theorem. Suppose is a directed-complete partial order (dcpo) with a least element, and let be a Scott-continuous (and therefore monotone) function. Then has a least fixed point, which is the supremum of the ascending Kleene chain of 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. Although Tarski's fixed point theorem does not consider how fixed points can be computed by iterating f from some seed (also, it pertains to monotone functions on complete lattices), this result is often attributed to Alfred Tarski who proves it for additive functions Moreover, Kleene Fixed-Point Theorem can be extended to monotone functions using transfinite iterations. Proof We first have to show that the ascending Kleene chain of exists in . To show that, we prove the following: Lemma. If is a dcpo with a least element, and is Scott-continuous, then Proof. We use induction: Assume n = 0. Then since is the least element. Assume n > 0. Then we have to show that . By rearranging we get . By inductive assumption, we know that holds, and because f is monotone (property of Scott-continuous functions), the result holds as well. As a corollary of the Lemma we have the following directed ω-chain: From the definition of a dcpo it follows that has a supremum, call it What remains now is to show that is the least fixed-point. First, we show that is a fixed point, i.e. that . Because is Scott-continuous, , that is . Also, since and because has no influence in determining the supremum we have: . It follows that , making a fixed-point of . The proof that is in fact the least fixed point can be done by showing that any element in is smaller than any fixed-point of (because by property of supremum, if all elements of a set are smaller than an element of then also is smaller than that same element of ). This is done by induction: Assume is some fixed-point of . We now prove by induction over that . The base of the induction obviously holds: since is the least element of