Consider the Hasse diagram of the lattice of subgroups of the quaternion group of order $8$ , $Q_8$ . [The use of $Q_8$ is only for a concrete realization of the lattice.]

To establish an order-preserving map which is not a lattice isomorphism one can simply ``skip'' $\langle -1\rangle$ , which we display graphically as:

Since containment is still preserved the map is order-preserving. However, the intersection (meet) of $\langle i\rangle$ and $\langle j\rangle$ , which is $\langle -1\rangle$ , is not perserved under this map. Thus it is not a lattice homomorphism.