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 $x_j \in S$.

We define the ordering $\preceq$ on $T_n$ as follows:


 * $\left({x_1, x_2, \ldots, x_n}\right) \prec \left({y_1, y_2, \ldots, y_n}\right)$ 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 $\displaystyle T = \bigcup_{n \ge 1} T_n$.

We define the ordering $\preceq$ on $T$ as follows:
 * $\left({x_1, x_2, \ldots, x_m}\right) \prec \left({y_1, y_2, \ldots, y_n}\right)$ 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 lexicographic (or lexicographical) order(ing).

Definition for Ordinals
The lexicographical ordering is a relation on ordered pairs of ordinals.

We shall denote it by $\operatorname{Le}$.

The lexicographical ordering is the collection of all ordered pairs $( ( \alpha, \beta ) , ( \gamma , \delta ) )$ such that:


 * Each $\alpha$, $\beta$, $\gamma$, $\delta$ is a member of the ordinal class.
 * $\alpha \in \gamma$ or $\alpha = \gamma \land \beta \in \delta$.

Also see

 * Finite Lexicographic Order on Well-Ordered Sets is Well-Ordering
 * Infinite Lexicographic Order on Well-Ordered Sets is Not Well-Ordering
 * Lexicographic Order Forms Well-Ordering on Ordered Pairs of Ordinals
 * Lexicographic Order Initial Segments (Result for ordinals)
 * Canonical Order, another strict well-ordering on $\operatorname{On}\times\operatorname{On}$