User:Dfeuer/Definition:Lexicographic Ordering on Product

Definition
Let $(I,\preceq)$ be a well-ordered set.

Let $(X_i,\le_i)$ be a totally ordered set for each $i \in I$.

Let $X = \prod_{i \mathop\in I} X_i$.

Let $M = \{k \in I \mid a_k \ne b_k \}$.

Define the relation $\le$ on $X$ as follows:

If $a,\,b \in X$, then $a \le b$ if and only if one of the following holds:


 * $M$ is empty.
 * $M$ is nonempty, and, letting $m = \min M$, and $a_m \le_m b_m$.

Then $\le$ is the lexicographic ordering on $X$.