# Definition:Order Product

## Definition

### Simple Order Product

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:
$\forall \tuple {a, b}, \tuple {c, d} \in T: \tuple {a, b} \preccurlyeq_s \tuple {c, d} \iff a \preccurlyeq_1 c \text { and } b \preccurlyeq_2 d$

### Lexicographic Order

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

The lexicographic order $\struct {S_1, \preccurlyeq_1} \otimes^l \struct {S_2, \preccurlyeq_2}$ on $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ is the ordered set $\struct {T, \preccurlyeq_l}$ where:

$T := S_1 \times S_2$, that is, the Cartesian product of $S_1$ and $S_2$
$\preccurlyeq_l$ is the relation defined on $T$ as:
$\tuple {x_1, x_2} \preccurlyeq_l \tuple {y_1, y_2} \iff \tuple {x_1 \prec_1 y_1} \lor \paren {x_1 = y_1 \land x_2 \preccurlyeq_2 y_2}$

### Antilexicographic Order

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

The antilexicographic order $\struct {S_1, \preccurlyeq_1} \otimes^a \struct {S_2, \preccurlyeq_2}$ on $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ is the ordered set $\struct {T, \preccurlyeq_a}$ where:

$T := S_1 \times S_2$, that is, the Cartesian product of $S_1$ and $S_2$
$\preccurlyeq_a$ is the relation defined on $T$ as:
$\tuple {x_1, x_2} \preccurlyeq_a \tuple {y_1, y_2} \iff \tuple {x_2 \prec_2 y_2} \lor \paren {x_2 = y_2 \land x_1 \preccurlyeq_1 y_1}$