How do the competing definitions for lattices relate?

Question

Definition 1: A lattice is an additive subgroup of $\mathbb{R}^n$ that is isomorphic to the additive group $\mathbb{Z}^m$ for some $m$.

Definition 2: A lattice is the set of all integer linear combinations of some set of basis vectors $\mathbf{b}_1,\dots,\mathbf{b}_n$: $$ \left\{\sum_i a_i\mathbf{b}_i\,\middle|\,a_i\in\mathbb{Z}\right\} $$

Definition 3: A lattice is a partially ordered set in which any two elements have a unique supremum and infimum.

Wikipedia also has other definitions that go straight over my head. My question is: which of these definitions are equivalent, which are special cases of the other, and why so?

Answer 1

The first two definitions are equivalente for lattice (in group theory) and the third definition is regarding lattices in order theory. Both are called lattice but are two different things. One kind are defined as discrete subgroups of R and the other kind are defined in terms of an order with a specific sup/inf opertion.

Answer 2

I'll add to Juan Ramirez answer the (sketch of the) proof that 2 is a particular case of 1.

Let's denote $L=\left\{\sum_i a_i\mathbf{b}_i\,\middle|\,a_i\in\mathbb{Z}\right\}$. Define the map $f:\mathbb{Z}^n\mapsto L$ by $f(a_1,\dots, a_n)=\sum_i a_i\mathbf{b}_i$. It has inverse $f^{-1}(\sum_i a_i\mathbf{b}_i)=(a_1,\dots, a_n)$. Here, the fact that $\{b_1,\dots, b_n\}$ is basis is important, since otherwise $f$ might not be injective (if they weren't linearly independent) or might not be surjective (if they don't generate $\mathbb{R}^n$).

It is clear that $f$ is a group homomorphism, since it preserves componentwise sum. Hence, both structures are isomorphic for $m=n$.

For the converse, there are some answers here.