Say we have $n$ variables $x_1,x_2...,x_n$.
Question 1.
We want to put conditions of the following form:
$$x_i+...+x_j>x_k+...+x_l$$ Where
The left side contains as many variables as the right side (but variables may be used multiple times, e.g.: $2\cdot x_3>x_2+x_1$), and
No single inequality is enough by itself to imply an inequality of the form $x_i>x_j$, i.e. with less than $3$ variables (this rules out, for instance, $x_1+x_2+x_3>x_1+x_2+x_4$).
We can have as many such inequalities as we want.
Is it possible for some $i,k$ to write down a set of such conditions such that all valuations that satisfy them also satisfy $x_i>x_k$?
Question 2.
What if we restrict to inequalities with certain coefficients? I.e. is it possible if we allow conditions of the form:
$$\alpha^1\cdot x_i+...+\alpha^{n}\cdot x_j>\alpha^1\cdot x_k+...+\alpha^n\cdot x_l$$
Where $\alpha\in (0,1)$, and $n$ is the number of variables in the inequality.
Is it possible in this case to induce $x_i>x_k$?
Doesn't something like $x_1+x_2>x_3+x_4$ and $x_3+x_4>x_2+x_5$ achieve what you're looking for?