# Category:Lexicographic Order

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This category contains results about **Lexicographic Order**.

Definitions specific to this category can be found in Definitions/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:
- $\tuple {x_1, x_2} \preccurlyeq_l \tuple {y_1, y_2} \iff \tuple {x_1 \prec_1 y_1} \lor \paren {x_1 = y_1 \land x_2 \preccurlyeq_2 y_2}$

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Lexicographic Order"

The following 13 pages are in this category, out of 13 total.

### I

### L

- Lexicographic Order forms Well-Ordering on Ordered Pairs of Ordinals
- Lexicographic Order is Ordering
- Lexicographic Order of Family of Totally Ordered Sets is Totally Ordered Set
- Lexicographic Order of Family of Well-Ordered Sets is not necessarily Well-Ordered
- Lexicographic Order on Pair of Totally Ordered Sets is Total Ordering
- Lexicographic Order on Pair of Well-Ordered Sets is Well-Ordering
- Lexicographic Order on Products of Well-Ordered Sets
- Lexicographic Product of Family of Ordered Sets is Ordered Set
- Lexicographically Ordered Pair of Ordered Semigroups with Cancellable Elements