by $$(a_1,a_2) \leq (b_1, b_2) \mbox{ if } a_i \leq_i b_i \mbox{ for } i = 1,2,$$ and the operations $\vee$ and $\wedge$ of the product are defined componentwise from the corresponding operations of the factors. Thus the product of distributive lattices is a distributive lattice. The [*(categorical) product*]{} of two graphs $G_1$ and $G_2$, is the graph $G = G_1 \times G_2$ with vertex set $V(G_1) \times V(G_2)$ and edgeset $$\{ (u_1,u_2)(v_1,v_2) \mid u_iv_i \in G_i \mbox{ for } i = 1,2\}.$$

The following is standard.

\[lem:product\] If the reflexive graph $G_i$ is compatible with the lattice $L_i$ for $i = 1,2$ then $G_1 \times G_2$ is compatible with $L_1 \times L_2$.

Let $(u_1,u_2)\sim (v_1,v_2)$ and $(u'_1,u'_2)\sim (v'_1,v'_2)$ in $G_1 \times G_2$. Then $$(u_1,u_2) \wedge (u'_1,u'_2) = (u_1 \wedge u'_1, u_2 \wedge u'_2) \sim
           (v_1 \wedge v'_1, v_2 \wedge v'_2) =  (v_1,v_2) \wedge (v'_1,v'_2),$$ and similarly $(u_1,u_2) \vee (u'_1,u'_2) =  (v_1,v_2) \vee (v'_1,v'_2)$.

A sublattice $L'$ of a lattice $L$ is any subset that is closed under the meet and join operations. The following is clear from the definition of compatibility.

\[fact:sublat\] If a graph $G$ is compatible with a lattice $L$, and $L'$ is a sublattice of $L$, 