Theorems for digraphs and lattices

Is this true: "all theorems that apply to all digraphs also apply to all lattices"

Short answer: No — not in any meaningful sense.

Here’s why (two perspectives):

1) The nitty-gritty logical view (technical “yes”). If you choose an encoding that turns every lattice $L$ into some directed graph $D(L)$ (for example the Hasse diagram or the directed comparability graph), then any statement of the form

$$ \forall G\in{\text{digraphs}}; P(G) $$

(which is true for every digraph $G$) is automatically true for those particular digraphs $D(L)$. So formally, a universally quantified digraph theorem holds for lattices viewed as those particular digraphs.

2) The practical / mathematical view (no). Most theorems in graph theory and most theorems in lattice theory live in different languages. Graph theorems talk about vertices, edges, indegree/outdegree, directed cycles, Euler tours, reachability, tournaments, etc. Lattice theorems talk about meets, joins, distributivity, modularity, ideals, congruences, etc. Many graph notions either:

• aren’t defined for a lattice as an algebraic object (what is “outdegree” of a lattice?); or
• don’t correspond to any natural lattice concept under the usual encodings; or
• use hypotheses (e.g. existence of cycles, strong connectivity, multiple edges, loops) that the standard digraph representation of a lattice (Hasse diagram) never satisfies.

So saying “all digraph theorems also apply to lattices” is misleading: a digraph theorem may be true for every digraph but be meaningless or irrelevant when you regard lattices as algebraic objects. Conversely, many lattice-theoretic theorems have no content in graph theory.

Concrete illustrations

• A graph-theory statement about Eulerian circuits or vertex degrees is meaningless when you take a lattice as a pure algebraic structure (there’s no built-in notion of “degree” in lattice theory).
• The Hasse diagram of a lattice is always acyclic (since it represents a partial order), so many digraph facts involving directed cycles simply never apply to Hasse diagrams — even though formally those Hasse diagrams are digraphs.

Bottom line: If you interpret every lattice as a particular kind of digraph, then logically a universal digraph theorem holds on those encodings. But in mathematical practice the two subjects use different languages and goals, so you cannot say that “all theorems about digraphs also hold as theorems about lattices” in any useful way.