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

Definition
Let $\struct {S, \preccurlyeq}$ be an ordered set.

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

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


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

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


 * $\tuple {x_1, x_2, \ldots, x_n} \preccurlyeq_l \tuple {y_1, y_2, \ldots, y_n}$ :
 * $\exists k: 1 \le k \le n: \paren {\forall j: 1 \le j < k: x_j = y_j} \land \paren {x_k \prec y_k}$
 * 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.