Fixed Point (mathematics) - Generalization To Partial Orders: Prefixpoint and Postfixpoint

Generalization To Partial Orders: Prefixpoint and Postfixpoint

The notion and terminology is generalized to a partial order. Let ≤ be a partial order over a set X and let f:X → X be a function over X. Then a prefixpoint (also spelled pre-fixpoint) of f is any p such that f(p) ≤ p. Analogously a postfixpoint (or post-fixpoint) of f is any p such that p ≤ f(p). One way to express the Knaster–Tarski theorem is to say that a monotone function on a complete lattice has a least fixpoint which coincides with its least prefixpoint (and similarly its greatest fixpoint coincides with its greatest postfixpoint). Prefixpoints and postfixpoints have applications in theoretical computer science.