It is standard to depict a poset by it’s Hasse diagram– its sub-digraph of covers– and to depict direction of the covers simply by assuming that the greater element is higher on the page. As we draw a lattice and a graph on the same set of vertices, the edges of our Hasse diagram are the thicker lighter edges, and the graph edges are thin and dark.

Recall that a [*lattice*]{} $L$ is a partial ordering on a set such that the greatest lower bound and least upper bound are uniquely defined for any pair of elements. These define the meet, $\wedge$, and join, $\vee$, operations respectively. It is a basic fact that the operations $\wedge$ and $\vee$ and the lattice defined each other by the identities $$u \leq v \iff (u \wedge v) = u \mbox{ and } u \leq v \iff (u \vee v ) = v.$$ The lattice is [*distributive*]{} if the meet and join distribute. As our lattices are finite the meet and join operations are well defined for any set of elements and there is a maximum element, or [*unit*]{}, denoted ${\mathbf{1}}$, and a minimum element or [*zero*]{} denoted ${\mathbf{0}}$.

A lattice $L$ on the 