Fixpoint

Fixed points of partial orders

In domain theory, the notion and terminology of fixed points 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 prefixed point (also spelled pre-fixed point, sometimes shortened to prefixpoint or pre-fixpoint) of f is any p such that f(p) ≤ p. Analogously, a postfixed point of f is any p such that p ≤ f(p). The opposite usage occasionally appears. Malkis justifies the definition presented here as follows: "since f is before the inequality sign in the term f(x) ≤ x, such x is called a prefix point." A fixed point is a point that is both a prefixpoint and a postfixpoint. Prefixpoints and postfixpoints have applications in theoretical computer science.