Let $X$ be a set. Let $0$ be an element of $X$.
For a function $P$ defined on tuples of $n$ elements of the set $X$ we know (for every tuples $f$ and $g$ each having $n$ elements) $$\forall i \in n : ( f_i \neq 0 \wedge g_i \neq 0 ) \wedge P f = P g \Rightarrow f = g.$$
Let $X$ be also a poset with least element $0$.
Under which additional conditions can we prove: $$\forall i \in n : ( f_i \neq 0 \wedge g_i \neq 0 ) \wedge P f \le P g \Rightarrow \forall i\in n: f_i \le g_i?$$
It is a practical task to prove it, don't be afraid to assume additional conditions. Maybe we should require that $X$ is a (semi)lattice?
There is no answer to this question. Consider for example when a reverse of the above formula holds: $$\forall i \in n : ( f_i \neq 0 \wedge g_i \neq 0 ) \wedge P f \le P g \Rightarrow \forall i\in n: f_i \ge g_i.$$
This case can't be distinguished from what is wanted in the question, without explicitly specifying a partial order for the image of $P$.