Are there any connections between these two definitions of a lattice?
- Let $B$ be a basis of $\mathbb R^n$. A lattice $L(B)$ is the set of all integer linear combinations of the vectors in the basis.
- A lattice is a poset $(L, \leq)$ such that any two-element subset of $L$ has a unique supremum and infimum.
For example, can all sets that are lattices according to the second definition be seen as "discretizations" (might not be the appropriate term here) of vector spaces on some field? (That is, is there a sense in which they also satisfy the other definition?)