Content deleted Content added
No edit summary |
No edit summary |
||
Line 4:
:''Let L be a [[complete partial order]], and let f : L → L be a (Scott-[[Scott continuity|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.
It is often attributed to [[Alfred Tarski]], but the original statement of [[Tarski's fixed point theorem]] is about monotone functions on complete lattices. Since all complete lattices are complete partial orders but not vice-versa (a complete lattice is a complete partial order with the additional property that every pair of lattice-elements has a supremum and an infimum in the lattice), and since all monotone functions on complete lattices are Scott-continuous, Tarski's fixed point theorem is
The '''ascending Kleene chain''' of ''f'' is the [[chain (order theory)|chain]]
| |||