Contents

# Contents

## Idea

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.

## Construction

###### Theorem

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:

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

## References

Named after Stephen Kleene.