N₅ — the pentagon

$N_5$ is the smallest non-modular lattice. It has five elements: a bottom $\bot$, a top $\top$, a chain $a < b$ on one side, and a single element $c$ on the other, incomparable with both $a$ and $b$. The modular law fails on the triple $(a, c, b)$:

$$a \vee (c \wedge b) = a \vee \bot = a, \quad (a \vee c) \wedge b = \top \wedge b = b.$$

Since $a < b$, modularity would force these to be equal — and they are not.

Where it appears

• $N_5$ is the subgroup lattice of $S_3$ (the symmetric group on three letters), restricted to its non-trivial proper subgroups together with the trivial and full groups.
• The Dedekind–Birkhoff theorem: a lattice is modular if and only if it has no $N_5$ sublattice; a modular lattice is distributive if and only if it has no $M_3$ sublattice.
• Pieces of $N_5$ show up whenever an order has “a shortcut” that bypasses an intermediate element.

If $M_3$ is the diamond, $N_5$ is its asymmetric cousin — and it is the structural reason modularity sometimes fails.