---
tag: 68WXFBJEA6
kind: definition
name: Direct product of algebras
msc: [08B25]
defines: WJ73XYBNE4
uses: [QJGPT2W0FT, 09CEDJYBKP]
sources:
  - url: https://en.wikipedia.org/wiki/Direct_product
---

## Definition

Let $I$ be a set and let $(\mathbf{A}_i)_{i \in I}$ be a family of [[algebras|QJGPT2W0FT]] of the same signature, with universes $A_i$.
The *direct product* $\prod_{i \in I} \mathbf{A}_i$ has as universe the [[Cartesian product|09CEDJYBKP]] $\prod_{i \in I} A_i$, the set of functions $a$ on $I$ with $a(i) \in A_i$ for each $i$.
Each $n$-ary fundamental operation $f$ is defined coordinatewise:
$$f(a_1, \dots, a_n)(i) = f^{\mathbf{A}_i}(a_1(i), \dots, a_n(i)) \quad \text{for all } i \in I.$$

## Remarks

For each $j \in I$ the projection $\pi_j \colon a \mapsto a(j)$ is a homomorphism onto $\mathbf{A}_j$ when the product is nonempty.
For two factors one writes $\mathbf{A}_1 \times \mathbf{A}_2$, with elements written as pairs.
The product of the empty family is a one-element algebra.

Example: the group $\Z/2\Z \times \Z/3\Z$ has six elements and is cyclic, generated by $(1, 1)$.
Coordinatewise, $k \cdot (1, 1) = (k \bmod 2, k \bmod 3)$, which is $(0, 0)$ first for $k = 6$.

Non-example: the group $\Z/4\Z$ is not isomorphic to the direct product $\Z/2\Z \times \Z/2\Z$, although both have four elements.
In the product, $x + x = 0$ for every $x$, since this holds in each coordinate, while $1 + 1 = 2 \ne 0$ in $\Z/4\Z$.

A direct product of algebras is the product in the category of algebras of the given signature, by the universal property of the projections.