- Kleene fixpoint theorem
In the mathematical areas of order and
lattice theory , the Kleene fixed-point theorem, named after American mathematicianStephen Cole Kleene , states the following::"Let L be acomplete partial order , and let f : L → L be a continuous (and therefore monotone) function. Then theleast fixed point of f is thesupremum of the ascending Kleene chain of f.It is often attributed to
Alfred Tarski , but the original statement ofTarski's fixed point theorem is about monotone functions on complete lattices.The ascending Kleene chain of "f" is the chain
:ot ; le ; f(ot) ; le ; fleft(f(ot) ight) ; le ; dots ; le ; f^n(ot) ; le ; dots
obtained by iterating "f" on the
least element ⊥ of "L". Expressed in a formula, the theorem states that:extrm{lfp}(f) = sup left(left{f^n(ot) mid ninmathbb{N} ight} ight)
where extrm{lfp} denotes the least fixed point.
See also
*
Knaster–Tarski theorem
* Otherfixed-point theorem s
Wikimedia Foundation. 2010.