Definition:Ordered Product

Definition
Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be tosets.

Let:
 * the order type of $\struct {S_1, \preceq_1}$ be $\theta_1$
 * the order type of $\struct {S_2, \preceq_2}$ be $\theta_2$.

Let $T = S_1 \times S_2$ be the cartesian product of $S_1$ and $S_2$.

Consider the relation $\preceq$ defined on $T$ as follows.

Let $a_1$ and $a_2$ be arbitrary elements of $S_1$, and $b_1$ and $b_2$ be arbitrary elements of $S_2$.

Then:
 * $b_1 \prec b_2 \implies \tuple {a_1, b_1} \prec \tuple {a_2, b_2}$
 * $b_1 = b_2, a_1 \prec a_2 \implies \tuple {a_1, b_1} \prec \tuple {a_2, b_2}$
 * $b_1 = b_2, a_1 = a_2 \implies \tuple {a_1, b_1} = \tuple {a_2, b_2}$

The ordered set $\struct {S_1 \times S_2, \preceq}$ is called the ordered product of $S_1$ and $S_2$, and is denoted $S_1 \cdot S_2$.

The order type of $S_1 \cdot S_2$ is denoted $\theta_1 \cdot \theta_2$.

General Definition
We can define the ordered product of any finite number of tosets as follows:

Also see

 * Antilexicographic Product of Totally Ordered Sets is Totally Ordered