Order-Preserving Function

Last updated Nov 1, 2022

Let $X, Y$ be Partial Orderings. Let $f: X \to Y$ be a Function. We say $f$ is order-preserving if for $x, x’ \in X$, $$x \leq x’ \Rightarrow f(x) \leq f(x’).$$

1. A Non-Decreasing Function is an Order-Preserving Function
2. An Order-Preserving Function may also be called a Monotone Function. I will avoid this though, because a Monotone Function may also refer to a Non-Increasing Function, which is not an Order-Preserving Function.