vertices of a graph $G$ was defined to be compatible if its meet and join operations $\wedge$ and $\vee$ are polymorphisms. Explicitly, $L$ is compatible with $G$ if and only if the following holds, where ‘$\sim$’ denotes adjacency in $G$: $$\label{id:poly}
    ( u \sim u' \mbox{ and }  v\sim v') \Rightarrow 
     (u \wedge v \sim u' \wedge v' \mbox{ and } u \vee v \sim u' \vee v').$$

Along with , there is another useful property of an ordering of vertices. $$\label{id:vee} 
   u \sim v \sim w  \mbox{ and } (u \leq v \geq w \mbox{  or } u \geq v \leq w)
   \Rightarrow u \sim w$$

\[fact:ids\] For a compatible lattice ordering of a reflexive graph $G$ identities and hold for all vertices $u,v,w$ of $G$.

For , as $v \sim v$ we get $u = u \wedge v \sim w \wedge v = v$ and $v = u \vee v \sim w \vee v = w$, as needed. For , assuming that $u \leq v \geq w$, we get $u = (u \wedge v) \sim (v \wedge w) = w$.

The product $L_1 \times L_2$ of two lattices is the ordering on the set $L_1 \times L_2$ defined 