Say there exists some lattice $L$ with a generator matrix $A$,
As well, there exists some lattice $J$ with a generator matrix $B$,
If $A=B$, is it necessarily the case that $L=J$?
I know that the opposite is not always true (i.e. as generator matrices are not unique, if $L=J$ then A and B are not necessarily equal as they could be many different options).
I suspect this is the case, but I have not been able to say with certainty. My thinking is that as $A$ is defined by a basis of $L$, then $A=B$ implies that $L$ and $J$ have the same basis and thus are the same, but I'm not sure my logic is sound.