Diamond lattice M3

Example

The diamond lattice $M_3$ has five elements $0, a, b, c, 1$. Its order is $0 < a < 1$, $0 < b < 1$ and $0 < c < 1$, with $a$, $b$, $c$ pairwise incomparable. The join of any two distinct elements among $a, b, c$ is $1$ and their meet is $0$, so $M_3$ is a lattice.

$M_3$ is a modular lattice. If $u \le v$ and $u = v$, or $u = 0$, or $v = 1$, the modular law $u \vee (x \wedge v) = (u \vee x) \wedge v$ reduces to a trivial identity; these cases cover all comparable pairs. For example, with $u = 0$ both sides equal $x \wedge v$, and with $v = 1$ both sides equal $u \vee x$.

$M_3$ is not a distributive lattice. Indeed $a \wedge (b \vee c) = a \wedge 1 = a$, while $(a \wedge b) \vee (a \wedge c) = 0 \vee 0 = 0$.