Monotone function

Definition

Let $(P, \le_P)$ and $(Q, \le_Q)$ be preorders and let $f \colon P \to Q$ be a function. The function $f$ is monotone when, for all $a, b \in P$, $a \le_P b$ implies $f(a) \le_Q f(b)$.

Remarks

Other names are order-preserving and, for real functions, monotonically increasing or non-decreasing. The function $f$ is antitone when $a \le b$ implies $f(b) \le f(a)$. The map $S \mapsto f(S)$ on power sets is monotone for $\subseteq$.

Explanations

L1 human

A monotone function never reverses the order of two inputs. On $\mathbb{R}$, $x \mapsto x^3$ is monotone, and $x \mapsto x^2$ is not, because $-2 \le 1$ but $4 > 1$.