A D18: Map $f : X \to Y$ is monotone from $P_X$ to $P_Y$ if and only if at least one of the following statements is true
(1) | $\forall \, x, y \in X \, (x \preceq_X y \quad \Rightarrow \quad f(x) \preceq_Y f(y))$ (D427: Isotone map) |
(2) | $\forall \, x, y \in X \, (x \preceq_X y \quad \Rightarrow \quad f(y) \preceq_Y f(x))$ (D428: Antitone map) |