What is it called when each element of one vector is greater than each element of another?

Question

I'm currently writing a term paper and while writing some proofs I've used an operator I don't know the name of.

Let us assume we have $x_1, x_2 \in \mathcal{R}^n$. Now lets define an operator '$\succeq$' such that $x_1 \succeq x_2$ implies that $x_1(i) \geq x_2(i)\,\,\forall\,\, i \in {1, .. n}$. Essentially, the operator implies that each element of one vector is greater than each element of the other. My question: is there a specific term for this operator? I strongly feel as though I've come across papers that have used this operator in proofs but I'm having a hard time finding them.

So far I've come across the concept of 'majorization' (http://mathworld.wolfram.com/Majorization.html) that seems to come close to what I want. Any help/information would be appreciated.

Answer 1

I don't really like a description of these as "vectors": I'd rather call them functions $\mathbb N\to \mathbb R$, or possibly with a domain consisting of some segment of $\mathbb N$.

Anyhow, in that context, I thought I had seen this called "the dominance order" on functions, which is a partial order you get when you say $f\leq g$ if $f(x)\leq g(x)$ for all $x$.

For example here or here.

However, while searching, I see that people use this term for lots of other partial orders that seem unrelated. Still, it seems like a pretty sensible name. You could say that one function(/vector) dominates another if it is greater at each point.

Answer 2

I've encountered the notation $x_1 \succeq x_2$ in the book Convex Optimization by Boyd and Vandenberghe. If $K$ is a cone, then they define the "generalized inequality" $x \succeq_K y$ to mean that $x - y \in K$. In your case, $K$ is the nonnegative orthant.