What is the mathematical structure called if we replace $\mathbb{R}^n$ by non negative integer $\mathbb{N}^n$ in hypercube?

Question

What is the mathematical structure called if we replace $\mathbb{R}^n$ by non negative integer $\mathbb{N}^n$ in hypercube? I am aware of Boolean hypercube which means each dimension could be $\{0,1\}$. What if we expand $\{0,1\}$ to all non negative integer $\mathbb{N}$? Is there a special name for this structure?

Answer 1

You can certainly speak about $\mathbb N^n$ if you have a use for it, but I'm not aware that it has a generally recognized fancy name of its own, like $\mathbb R^n$ is known as "euclidean space".

In abstract algebra, $\mathbb N^n$ with (elementwise) addition is the free commutative monoid with $n$ generators, but I daresay most readers would take longer to absorb that description than they would to recognize $\mathbb N^n$.