Lexicographic order

Definition

Let $(A, \le_A)$ and $(B, \le_B)$ be partially ordered sets. The lexicographic order on the Cartesian product $A \times B$ is the relation $\le$ with $(a, b) \le (a', b')$ when $$a <_A a' \quad \text{or} \quad (a = a' \text{ and } b \le_B b').$$

Remarks

It is a partial order. When $\le_A$ and $\le_B$ are total orders, it is a total order. Example: in $\mathbb{N} \times \mathbb{N}$, $(1, 5) < (2, 0)$ and $(2, 0) < (2, 3)$. A dictionary orders words in the same way, letter by letter.

Explanations

L1 human

The lexicographic order is the dictionary order. "cat" comes before "cow", because the first letters agree and "a" comes before "o".