Definition:Lexicographic Order/General Definition

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

For $n \in \N: n > 0$, we define $T_n$ as the set of all ordered $n$-tuples:
 * $\tuple {x_1, x_2, \ldots, x_n}$

of elements $x_j \in S$.

Let $\displaystyle T = \bigcup_{n \mathop \ge 1} T_n$.

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


 * $\tuple {x_1, x_2, \ldots, x_m} \preccurlyeq \tuple {y_1, y_2, \ldots, y_n}$ :
 * $\exists k: 1 \le k \le \map \min {m, n}: \paren {\forall j: 1 \le j < k: x_j = y_j} \land x_k \prec y_k$
 * or:
 * $m \le n$ and $\forall j: 1 \le j \le m: x_j = y_j$.

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

or:
 * all elements are equal up to the length of the shorter one.

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