Definition:Lexicographic Order

Let $$S$$ be a set which is well-ordered by $$\preceq$$.

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, we define $$T = \bigcup_{n \ge 1} T_n$$.

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 order.

It can be shown that $\preceq$ is not a well-ordering on $T$, but that $$\preceq$$ is a well-ordering on a given $$T_n$$.