nt.number theory - Q-lattices and commensurability, any insight into the definition and intuition?

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.