I've been coming across $\mathbb{Q}$-lattices in $\mathbb{R}^n$ in my reading, and I'm having trouble understanding the definitions. Connes and Marcolli define it as a lattice $\Lambda \in \mathbb{R}^n$ together with a homomorphism $\phi : \mathbb{Q}^n / \mathbb{Z}^n \to \mathbb{Q} \Lambda / \Lambda$. Moreover, two $\mathbb{Q}$-lattices $\Lambda_1$ and $\Lambda_2$ are commensurable iff 1) $\mathbb{Q} \Lambda_1 = \mathbb{Q}\Lambda_2$ and 2) $\phi_1 = \phi_2$ mod $\Lambda_1 + \Lambda_2$.

I think I understand condition 1): the lattices must be rational multiples of each other to be commensurable. I don't even understand the notation for condition 2). The best I can gather is that the homomorphism $\phi$ labels which positions in $\mathbb{Q} \Lambda / \Lambda$ come from your more normal "discrete hyper-torus" $\mathbb{Q}^n / \mathbb{Z}^n$. Condition 2) then says that the same points are labelled. Is this anywhere near the right ballpark? Can anyone recommend any literature on the subject?

I'm a pretty young mathematician (not even in a PhD program...yet) so please forgive me if this question seems basic.

Thanks.