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In the [[mathematics|mathematical]] areas of [[order theory|order]] and [[lattice theory]], the '''Kleene fixed-point theorem''', named after American mathematician [[Stephen Cole Kleene]], states the following:
:''Let (L, ⊑) be a CPO ([[complete partial order]]), and let f : L → L be a [[Scott continuity|Scott-continuous]] (and therefore [[monotone function|monotone]]) [[function (mathematics)|function]]. Then f has a [[least fixed point]], which is the [[supremum]] of the ascending Kleene chain of f.
The '''ascending Kleene chain''' of ''f'' is the [[chain (order theory)|chain]]
:<math>\bot \; \
obtained by [[iterated function|iterating]] ''f'' on the [[least element]] ⊥ of ''L''. Expressed in a formula, the theorem states that
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We first have to show that the ascending Kleene chain of ''f'' exists in L. To show that, we prove the following lemma:
:Lemma 1:''If L is CPO, and f : L → L is a Scott-continuous, then <math>f^n(\bot) \
Proof by induction:
* Assume n = 0. Then <math>f^0(\bot) = \bot \
* Assume n > 0. Then we have to show that <math>f^n(\bot) \
Immediate corollary of Lemma 1 is the existence of the chain.
Let <math>\mathbb{M}</math> be the set of all elements of the chain: <math>\mathbb{M} = \{ \bot, f(\bot), f(f(\bot)), \ldots\}</math>. This set is clearly a directed/ω-chain, as a
First, we show that <math>m</math> is a fixed point, i.e. that <math>f(m) = m</math>. Because <math>f</math> is [[Scott continuity|Scott-continuous]], <math>f(\sup(\mathbb{M})) = \sup(f(\mathbb{M}))</math>, that is <math>f(m) = \sup(f(\mathbb{M}))</math>. Also, since <math>f(\mathbb{M}) = \mathbb{M}\setminus\{\bot\}</math> and because <math>\bot</math> has no influence in determining <math>\sup</math>, we have that <math>\sup(f(\mathbb{M})) = \sup(\mathbb{M})</math>. It follows that <math>f(m) = m</math>, making <math>m</math> a fixed-point of <math>f</math>.
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