I am trying to define an objective function for my optimization problem and I came up with several alternatives, while it is not clear, which one is the most relevant.
But I think it would be nice if there was some sort of consistency relationship between these functions. For example, if $f,g: X \mapsto \mathbb{R} $ are two objective functions, then for each $x,y \in X$ I would like to have:
$$ f(x) < f(y) \implies g(x) < g(y) $$
I am surely not the first one to think in such a manner, so I would like to ask if there is some special name for this relationship between functions.
The closest I got was by reading section 2 of this paper, where they define a so called relation refinement. That means, that the relation of total preoder induced by $g$ is a refinement of the one induced by $f$. However, I couldn't find any definition dealing with functions $f,g$ themselves.