We are given a rank $r$ matrix $B\in\Bbb Z^{k\times n}$ where $0\leq r\leq k\leq n$ holds.


We have $\mathcal L\_\Bbb Z\subseteq \mathcal L\_\Bbb Q\subseteq\mathcal L\_\Bbb R$ where 
$$\mathcal L\_\Bbb Z=\{uB\in\Bbb Z^n:u\in\Bbb Z^k\},\quad\mathcal L\_\Bbb Q=\{uB\in\Bbb Z^n:u\in\Bbb Q^k\},\quad\mathcal L\_\Bbb R=\{uB\in\Bbb Z^n:u\in\Bbb R^k\}$$ holds.


When can we have


1. $\mathcal L\_\Bbb Z\subsetneq \mathcal L\_\Bbb Q\subsetneq\mathcal L\_\Bbb R$?
2. $\mathcal L\_\Bbb Z\subsetneq \mathcal L\_\Bbb Q=\mathcal L\_\Bbb R$?
3. $\mathcal L\_\Bbb Z= \mathcal L\_\Bbb Q\subsetneq\mathcal L\_\Bbb R$?
4. $\mathcal L\_\Bbb Z= \mathcal L\_\Bbb Q=\mathcal L\_\Bbb R$?



Integer points spanned by real, rational and integer combination of integer vectors
We always have $\mathcal L\_\Bbb Q=\mathcal L\_\Bbb R:$ For any particular $v \in\Bbb Z^n$ consider how we determine $\{u \in \Bbb R^n \mid uB=v\}.$ The method, if the set is nonempty, will yield one or more solutions with $u \in \Bbb Q^n.$


The Smith Normal Form of $B$ is a certain integer diagonal matrix. I'd expect that we have $\mathcal L\_\Bbb Z\subsetneq \mathcal L\_\Bbb Q$ exactly when this matrix has a diagonal entry greater than $1.$ 


