Category:Ordered Products

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This category contains results about Ordered Products.


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 $S_1 \cdot S_2$.

Pages in category "Ordered Products"

The following 2 pages are in this category, out of 2 total.