How should we notate a D-dimensional lattice defined on multiples of natural numbers? I have seen the lattice of natural numbers notated as $\mathrm{L}_\mathbb{N}$.
It makes sense to me to write the $D$-dimensional lattice of natural numbers $\mathrm{L}_{\mathbb{N}^D}$. However, I’m not a mathematician, much less an expert in lattices, so I’m writing to ask if this is correct? And, if not, how should I notate this lattice?
I also need to notate a lattice defined on multiples of natural numbers, e.g., $\{0, 2, 4, \dots \}$. How should that be written?
 A: It's fairly common to just write $\mathbb N^D$ for the lattice* of natural numbers - the meaning of this just comes from the more general meaning that $S^D$ means** "the set of tuples of length $D$ with values in $S$"- so $\mathbb N^D$ means "the set of tuples of $D$ natural numbers" which is the lattice you are referring to.
It is also common to, wherever scalar multiplication is defined, to write $kS$ to mean the set of elements of the form $ks$ where $s\in S$; for instance, $2\mathbb Z$ is the set of even integers in this notation.
You can put these together to write $2\mathbb N^D$ to mean the set of lattice points in $D$ dimensions whose coordinates are all even. This is clear and concise once understtood. As with almost all notation, it'd be worth explaining in words what this means if you used it in writing - it's common notation in certain fields (e.g. when discussing modules) and most mathematicians could guess the meaning of it in context, but it's almost always a good idea to explain your notation;

*This isn't actually a lattice in the usual sense, on account of it not being closed under negation, but close enough.
**More generally $S^D$ means "the set of functions from $D$ to $N$" but when $D$ is a finite number, this is generally understood as a set of tuples; of course "function on a finite set" and "tuple" turn out to be the same thing.
