How best to explain partial order over vectors?

Question

I have the following notational issue:

Given a vector space $\mathbf{V}$, I have a partial order such that for all $\mathbf{v},\mathbf{w} \in \mathbf{V}$:

iff for all components $v_i$ of $\mathbf{v}$ and $w_i$ of $\mathbf{w}$ $$v_i\leq w_i$$

then

$$\mathbf{v} \leq \mathbf{w}$$

Is there a well known way of stating this partial order ?

Answer 1

I had not heard of this concept before, but searching for "componentwise partial order" brought up the "product order", which seems to be what you want.

I'm not an order theorist, but if you didn't know of a name for this offhand, your audience may not know this definition of "product order", so it would be good to give the definition in whatever you are writing, anyway.