Dedekind's characterization of modular lattices

Remarks

One direction is easy: $N_5$ is not modular, and the modular law holds in every sublattice of a modular lattice. For the other direction, suppose $a \le b$ and $x$ violate the law, and put $u = a \vee (x \wedge b)$ and $v = (a \vee x) \wedge b$, so $u < v$. A direct computation gives $x \wedge u = x \wedge v = x \wedge b$ and $x \vee u = x \vee v = a \vee x$. So the five elements $x \wedge b$, $u$, $v$, $x$ and $a \vee x$ form a sublattice isomorphic to $N_5$.

Example: the lattice of subspaces of a vector space is modular, so it contains no copy of $N_5$. Together with the M3–N5 characterization of distributivity, it shows that the pentagon and the diamond are the only obstructions to distributivity.

Going deeper

For a student

The theorem turns an equational condition into a forbidden-substructure condition. It is used to show that a given lattice is not modular by exhibiting a pentagon, and to prove modularity of lattices built from modular pieces. The same style of result, with the diamond $M_3$ added, characterizes distributive lattices. The proof is short and gives the pentagon explicitly from any failure of the modular law.