*Kleene’s fixed point theorem* theorem constructs least fixed points of endofunctions on posets by iterating them. Adámek's fixed point theorem generalizes this to constructing initial algebras.

Let $f : P \to P$ be a monotone function on a poset $P$. If $P$ has a least element $\bot$ and joins of increasing sequences, and if $f$ preserves joins of increasing sequences, then a least fixed point of $f$ can be constructed as the join of the increasing sequence:

$\bot
\;\leq\;
f(\bot)
\;\leq\;
f^2(\bot)
\;\leq\;
\cdots
\,.$

Named after Stephen Kleene.

See also:

- Wikipedia,
*Kleene fixed point theorem*

Last revised on November 29, 2023 at 16:41:41. See the history of this page for a list of all contributions to it.