# Category:External Direct Products

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This category contains results about **External Direct Products**.

Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be algebraic structures.

The **(external) direct product** $\struct {S \times T, \circ}$ of $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ is the set of ordered pairs:

- $\struct {S \times T, \circ} = \set {\tuple {s, t}: s \in S, t \in T}$

where the operation $\circ$ is defined as:

- $\tuple {s_1, t_1} \circ \tuple {s_2, t_2} = \tuple {s_1 \circ_1 s_2, t_1 \circ_2 t_2}$

## Subcategories

This category has the following 6 subcategories, out of 6 total.

### E

### G

## Pages in category "External Direct Products"

The following 23 pages are in this category, out of 23 total.

### C

### E

- Equivalent Conditions for Entropic Structure/Mapping from External Direct Product is Homomorphism
- Equivalent Conditions for Entropic Structure/Pointwise Operation of Homomorphisms from External Direct Product is Homomorphism
- External Direct Product Associativity
- External Direct Product Closure
- External Direct Product Commutativity
- External Direct Product Identity
- External Direct Product Inverses
- External Direct Product of Congruence Relations
- External Direct Product of Groups is Group
- External Direct Product of Projection with Canonical Injection
- External Direct Product of Projection with Canonical Injection/General Result
- External Direct Product of Ringoids is Ringoid
- External Direct Product of Semigroups