I have started to study the theory of orders in quaternion algebras, but I am very new to the topic. In particular I am interested in the Class set associated with an order in such an algebra. I have been trying to make parallels with classical algebraic number theory which I am much more familiar with, and to see what changes and what stays the same in this non-commutative setting.
In particular I am trying to get intuition for why the compatibility of lattices is an interesting notion. I understand how products of compatible lattices $I$ and $J$ are a special case of the tensor-products of $\mathcal{O}$-modules, where $\mathcal{O} = \mathcal{O}_l(J) = \mathcal{O}_R(I)$, as the map $i \otimes j \mapsto ij$ is an isomorphism of $\mathcal{O}$-modules in this setting. In the end though, even when considering only products of compatible lattices, one fails to obtain a group structure on the ideal class set $Cls \mathcal{O}$. Even in the nice case where $\mathcal{O}$ is maximal, so that every fractional (say left) $\mathcal{O}$-ideal is invertible things do not work out. For this reason, I fail to understand why one should bother to restrict the attention to compatible lattices in the first place. Any kind of insight would be greatly appreciated.