# Definition:Simple Order Product

## Definition

Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.

The simple (order) product $\struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$ of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ is the ordered set $\struct {T, \preccurlyeq_s}$ where:

$T := S_1 \times S_2$, that is, the Cartesian product of $S_1$ and $S_2$
$\preccurlyeq_s$ is defined as:
$\forall \tuple {a, b}, \tuple {c, d} \in T: \tuple {a, b} \preccurlyeq_s \tuple {c, d} \iff a \preccurlyeq_1 c \text { and } b \preccurlyeq_2 d$

### Family of Ordered Sets

Let $I$ be an indexing set.

For each $i \in I$, let $\struct {S_i, \preccurlyeq_i}$ be an ordered set.

Let $\ds D = \prod_{i \mathop \in I} S_i$ be the Cartesian product of the family $\family {\struct {S_i, \preccurlyeq_i} }_{i \mathop \in I}$ indexed by $I$.

Then the simple order product on $D$ is defined as:

$\ds \struct {D, \preccurlyeq_D} := {\bigotimes_{i \mathop \in I} }^s \struct {S_i, \preccurlyeq_i}$

where $\preccurlyeq_D$ is defined as:

$\forall u, v \in D: u \preccurlyeq_D v \iff \forall i \in I: \map u i \preccurlyeq_i \map v i$

## Also known as

Expositions which do not analyse the various standard order types on a Cartesian product can be seen to refer to this concept merely as the Cartesian product of ordered sets.

## Examples

### Unit Square with Open Side

Consider the simple order product of the real intervals $\hointr 0 1$ and $\closedint 0 1$ under the usual ordering:

$\struct {T, \preccurlyeq_s} := \struct {\hointr 0 1, \le} \otimes^s \struct {\closedint 0 1, \le}$

$\struct {T, \preccurlyeq_s}$ has one minimal element:

$\tuple {0, 0}$

which is also the smallest element: of $\struct {T, \preccurlyeq_s}$.

$\struct {T, \preccurlyeq_s}$ has no greatest element and no maximal elements.

## Also see

• Results about simple order product can be found here.