\begin{definition}[Definition:Lexicographic Order/General Definition]
Let $\struct {S, \preccurlyeq}$ 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 $\ds T = \bigcup_{n \mathop \ge 1} T_n$.
The '''lexicographic order on $T$''' is the relation $\preccurlyeq_l$ defined on $T$ as:
:$\tuple {x_1, x_2, \ldots, x_m} \preccurlyeq_l \tuple {y_1, y_2, \ldots, y_n}$ {{iff}}:
::$\exists k: 1 \le k \le \map \min {m, n}: \paren {\forall j: 1 \le j < k: x_j = y_j} \land x_k \prec y_k$
:or:
::$m \le n$ and $\forall j: 1 \le j \le m: x_j = y_j$.
That is, {{iff}}:
: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.
\end{definition}