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:

Kleene Fixed-Point Theorem. Suppose ${\displaystyle (L,\sqsubseteq )}$ is a directed-complete partial order (dcpo) with a least element, and let ${\displaystyle f:L\to L}$ be a Scott-continuous (and therefore monotone) function. Then ${\displaystyle f}$ has a least fixed point, which is the supremum of the ascending Kleene chain of ${\displaystyle f.}$

The ascending Kleene chain of f is the chain

${\displaystyle \bot \sqsubseteq f(\bot )\sqsubseteq f(f(\bot ))\sqsubseteq \cdots \sqsubseteq f^{n}(\bot )\sqsubseteq \cdots }$

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

${\displaystyle {\textrm {lfp}}(f)=\sup \left(\left\{f^{n}(\bot )\mid n\in \mathbb {N} \right\}\right)}$

where ${\displaystyle {\textrm {lfp}}}$ 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.