Can inequalities over $n>2$ variables ever imply an inequality over $2$ variables?

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

1. 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

2. 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 $$x_i>x_k$$?

• You need another restriction: that no variable appears on both sides of an equation. Otherwise you could have $x_1+x_2>x_1+x_3$, forcing $x_2>x_3$. – TonyK Nov 4 '18 at 12:54
• @TonyK, thanks, edited. – user600670 Nov 4 '18 at 12:59
• The way you put it is guaranteed to have readers scratching their heads... – TonyK Nov 4 '18 at 13:04
• ...but I suppose it does also rule out other undesirable equations, like $2x_1>2x_2$. – TonyK Nov 4 '18 at 13:13
• @TonyK, is this better? – user600670 Nov 4 '18 at 16:01

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?