Definition:Lexicographic Order

Definition
Let $\left({S, \preceq}\right)$ be a poset.

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

of elements $x_j \in S$.

The ordering $\preceq$ is defined on $T_n$ as:


 * $\left({x_1, x_2, \ldots, x_n}\right) \prec \left({y_1, y_2, \ldots, y_n}\right)$ iff:
 * $\exists k: 1 \le k \le n$ such that $\forall 1 \le j < k: x_j = y_j$ but $x_k \prec y_k$ in $S$.

Next, let $\displaystyle T = \bigcup_{n \mathop \ge 1} T_n$.

The ordering $\preceq$ is defined on $T$ as:


 * $\left({x_1, x_2, \ldots, x_m}\right) \prec \left({y_1, y_2, \ldots, y_n}\right)$ iff:
 * $\exists k: 1 \le k \le \min \left({m, n}\right)$ such that $\forall 1 \le j < k: x_j = y_j$ but $x_k \prec y_k$ in $S$
 * or:
 * $m < n$ and $\forall 1 \le j < m: x_j = y_j$.

This ordering is called (the) lexicographic order.

Definition for Ordinals
The lexicographic order is a relation on ordered pairs of ordinals denoted $\operatorname{Le}$.

$\operatorname{Le}$ is the set of all ordered pairs $\left({\left({\alpha, \beta}\right), \left({\gamma, \delta}\right)}\right)$ such that:


 * $(1): \quad$ Each $\alpha$, $\beta$, $\gamma$, $\delta$ is a member of the ordinal class
 * $(2): \quad$ $\alpha \in \gamma$ or $\alpha = \gamma \land \beta \in \delta$

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

Also see

 * Definition:Canonical Order