# From an equality to a comparison

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?

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You seemed to have a little trouble with your english so I corrected things a little. –  Patrick Da Silva Jul 7 '12 at 16:29
Perhaps that if you define $X^n$ as a poset by letting $f \le g$ if $f_i \le g_i$ for $i \in n$, this would be equivalent to prove that $Pf \le Pg \quad f \le g$, and would make things a little clearer. But why do you want to prove such a thing? Is there any intuition on why you need this? I think there is but I don't see it right away. –  Patrick Da Silva Jul 7 '12 at 16:31
@PatrickDaSilva: I want to prove that my generalized continuity for a product of morphisms implies (except of the case when one of the morphisms is zero) generalized continuity of each multiplier. Currently it is on the pages 29-30 after the theorem 159 of mathematics21.org/binaries/nary.pdf –  porton Jul 7 '12 at 16:36
Ugh, it seems that there are no simple condition for this (and my question has no answer). Consider for example the case when the orders are dual (reverse) to the formula above. Then the formula above, which I want, does not hold, but it seems to not contradict to any essential condition we may want to specify. –  porton Jul 7 '12 at 16:48

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 $P$.