M₃ — the diamond

$M_3$ is the five-element lattice with a bottom $\bot$, a top $\top$, and three mutually incomparable atoms $a, b, c$, each of which complements the other two. It is modular — Birkhoff’s theorem says a lattice is modular if and only if it contains no sublattice isomorphic to $N_5$ — yet it is not distributive, because $a \wedge (b \vee c) = a \wedge \top = a \ne \bot = (a \wedge b) \vee (a \wedge c).$

Where it appears

• The subgroup lattice of $\mathbb{Z}_2 \times \mathbb{Z}_2$ is $M_3$.
• Any non-distributive modular lattice contains $M_3$ as a sublattice (the ”$M_3$ – $N_5$ theorem” of Birkhoff and Dedekind).
• Projective geometries of low dimension produce $M_3$ as the lattice of subspaces of a 2-dimensional space over $\mathbb{F}_2$.

$M_3$ is the smallest witness that modularity is a strictly weaker condition than distributivity.