Definition:Lexicographic Order/Tuples of Equal Length/Cartesian Space

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $n \in \N_{>0}$.

Let $S^n$ be the cartesian $n$th power of $S$:


 * $S^n = \underbrace{S \times S \times \cdots \times S}_{n \text{ times} }$

The lexicographic order on $S^n$ is the relation $\preccurlyeq$ defined on $S^n$ as:


 * $\left({x_1, x_2, \ldots, x_n}\right) \preccurlyeq \left({y_1, y_2, \ldots, y_n}\right)$ :
 * $\exists k: 1 \le k \le n: \left({\forall j: 1 \le j < k: x_j = y_j}\right) \land \left({x_k \prec y_k}\right)$
 * or:
 * $\forall j: 1 \le j \le n: x_j = y_j$

That is, :
 * the elements of a pair of $n$-tuples are either all equal

or:
 * they are all equal up to a certain point, and on the next one they are comparable and they are different.

Also known as
Lexicographic order can also be known as the more unwieldy lexicographical ordering.