is the intersection with a Lattice still a Lattice?

Given a lattice A in $R^n$, and a subspace B of $R^n$. is the intersection $A \cap B$ a lattice?

Thanks

Well, is $A\cap B$ a discrete set of points in a Euclidean space? Yes. Is it closed under addition and taking opposite vectors? Also yes. Thus, it is a lattice. It may not have full rank in $B$ though; in particular, it may have rank $0$ (i. e. consist only of the origin).

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