Definition:Lexicographic Order/Tuples of Equal Length

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

Let $\left({S_1, \preceq_1}\right), \left({S_2, \preceq_2}\right), \ldots, \left({S_n, \preceq_n}\right)$ be ordered sets.

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

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


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

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

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