# Category:Order Sums

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

Definitions specific to this category can be found in Definitions/Order Sums.

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

The **order sum** $\struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2}$ of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ is the ordered set $\struct {T, \preccurlyeq}$ where:

- $T := S_1 \sqcup S_2 = \paren {S_1 \times \set 0} \cup \paren {S_2 \times \set 1}$
- $\preccurlyeq$ is defined as:
- $\forall \tuple {a, b}, \tuple {c, d} \in T: \tuple {a, b} \preccurlyeq \tuple {c, d} \iff \begin {cases} b = 0 \text { and } d = 1 \\ b = d = 0 \text { and } a \preccurlyeq_1 c \\ b = d = 1 \text { and } a \preccurlyeq_2 c \end {cases}$

## Subcategories

This category has only the following subcategory.

### E

## Pages in category "Order Sums"

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

### O

- Order Isomorphism is Preserved by Order Sum
- Order Sum of Ordered Sets is Ordered
- Order Sum of Totally Ordered Sets is Totally Ordered
- Order Sum of Well-Founded Orderings is Well-Founded Ordering
- Order Type Addition is Associative
- Order Type Addition is Well-Defined Operation
- Order Type Multiplication Distributes over Addition