Bourbaki–Witt theorem

In mathematics, the Bourbaki–Witt theorem in order theory, named after Nicolas Bourbaki and Ernst Witt, is a basic fixed point theorem for partially ordered sets. It states that if X is a non-empty chain complete poset, and ${\displaystyle f:X\to X}$ such that ${\displaystyle f(x)\geq x}$ for all ${\displaystyle x,}$ then f has a fixed point. Such a function f is called inflationary or progressive.