# Category:Simple Order Product

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This category contains results about **Simple Order Product**.

Definitions specific to this category can be found in Definitions/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:
- $\forall \tuple {a, b}, \tuple {c, d} \in T: \tuple {a, b} \preccurlyeq_s \tuple {c, d} \iff a \preccurlyeq_1 c \text { and } b \preccurlyeq_2 d$

## Pages in category "Simple Order Product"

The following 7 pages are in this category, out of 7 total.

### S

- Simple Order Product of Pair of Ordered Semigroups is Ordered Semigroup
- Simple Order Product of Pair of Ordered Sets is Lattice iff Factors are Lattices
- Simple Order Product of Pair of Ordered Sets is Ordered Set
- Simple Order Product of Pair of Totally Ordered Sets is Total iff One Factor is Singleton
- Simple Order Product of Totally Ordered Sets may not be Totally Ordered
- Supremum by Suprema of Directed Set in Simple Order Product
- Supremum of Simple Order Product