Distributive lattice

Definition

A lattice $(L, \vee, \wedge)$ is distributive if $x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z)$ for all $x, y, z \in L$.

Remarks

The condition is equivalent to its dual: $x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z)$ for all $x, y, z$. Every totally ordered set is a distributive lattice, with $\max$ as join and $\min$ as meet. Every Boolean algebra is a distributive lattice. The diamond lattice $M_3$ and the pentagon lattice $N_5$ are the basic lattices that are not distributive.

Explanations

L1 human

A lattice is distributive when meet distributes over join, like intersection over union for sets. Then join also distributes over meet.