Definition:Lexicographic Order/Tuples of Equal Length

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

Let $\struct {S_1, \preccurlyeq_1}, \struct {S_2, \preccurlyeq_2}, \ldots, \struct {S_n, \preccurlyeq_n}$ be ordered sets.

Let $\ds S = \prod_{k \mathop = 1}^n S_k = S_1 \times S_2 \times \cdots \times S_n$ be the Cartesian product of $S_1$ to $S_n$.

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


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

Cartesian Space
The definition can be refined to apply to a Cartesian $n$-space: