As I roughly know, a function $f$ in $\mathbb{R}$ is called order-preserving if $f(x)>f(y)$ for $x>y$, $x,y \in \mathbb{R}$. May I ask, if anybody knows the terminology for two functions $f$ and $g$ (or what $g$ is to $f$) if $g(x)>g(y)$ for $f(x)>f(y)$ ?
Also, the terminology when $g(x)>g(y)$ for $f(x)<f(y)$? Many thanks :)
I don't know of any standard terminology, but one option would be to define that whenever $P$ is a poset and $f : X \rightarrow P$ is a function, we have $$A(f) = \{(x,y) \in X^2 \mid f(x) < f(y)\}.$$
Then the relation of interest between $f$ and $g$ can be denoted $A(f) \subseteq A(g),$ which can be abbreviated $f \subseteq g : A$ (read: $f$ is a subset of $g$ when viewed under $A$).
By the way, a function that preserves $\leq$ is typically called order-preserving. A function that preserves $<$ might be called "strict-order preserving."