On page 8 of this pdf (https://www.lix.polytechnique.fr/~vjost/mpri/prelimILP.pdf) from Schrijvers book "combinatorial optimization" the dominant of a polytope is defined as:
P$^\uparrow$ := {$y \in \mathbb{R}^n | \exists x \in P : y \geq x$ } = P + $\mathbb{R}^n_+$
P$^\downarrow$ := {$y \in \mathbb{R}^n | \exists x \in P : y \leq x$ } = P - $\mathbb{R}^n_+$
for P $\subseteq \mathbb{R}^n$
how can $y$ be greater or less than $x$ when both are in $\mathbb{R}^n$. how can they be compared?
And what do P + $\mathbb{R}^n_+$ and P - $\mathbb{R}^n_+$ mean?