Definition:Simple Order Product

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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.


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.