nLab Kleene's fixed point theorem

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:

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

Related entries

• fixed point

• Adámek's fixed point theorem

• Knaster-Tarski's fixed point theorem

• Lawvere's fixed point theorem

• Lefschetz fixed point theorem

• Brouwer's fixed point theorem

• Atiyah-Bott fixed point formula

References

Named after Stephen Kleene.

See also:

• Wikipedia, Kleene fixed point theorem