Least fixed point

Summary

The least fixed point of a map $f$ from a partially ordered set to itself is a point $x$ with $f(x) = x$. It must lie below every other point with this property. When it exists it is unique. Least fixed points give meaning to recursive and inductive definitions: the set defined inductively by some rules is the least set closed under the rules. The Knaster–Tarski theorem and the Kleene fixed-point theorem guarantee least fixed points of monotone maps under completeness hypotheses.