Definition:Ordered Product

Definition
Let $$\left({S_1, \preceq_1}\right)$$ and $$\left({S_2, \preceq_2}\right)$$ be tosets.

Let:
 * the order type of $$\left({S_1, \preceq_1}\right)$$ be $$\theta_1$$;
 * the order type of $$\left({S_2, \preceq_2}\right)$$ 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:
 * $$b_1 \prec b_2 \implies \left({a_1, b_1}\right) \prec \left({a_2, b_2}\right)$$
 * $$b_1 = b_2, a_1 \prec a_2 \implies \left({a_1, b_1}\right) \prec \left({a_2, b_2}\right)$$
 * $$b_1 = b_2, a_1 = a_2 \implies \left({a_1, b_1}\right) = \left({a_2, b_2}\right)$$

The ordered set $$\left({S_1 \times S_2, \preceq}\right)$$ 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 $$\theta_1 \cdot \theta_2$$.

The Ordered Product of Tosets is a Totally Ordered Set.

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

Let $$S_1, S_2, \ldots, S_n$$ all be tosets.

Then we define $$T_n$$ as the ordered product of $$S_1, S_2, \ldots, S_n$$ as:


 * $$\forall n \in \N^*: T_n = \begin{cases}

S_1 & : n = 1 \\ T_{n-1} \cdot S_n & : n > 1 \end{cases}$$

Note
The ordered product is defined only for totally ordered sets.