Direct product of algebras

Builds on Algebraic structure Cartesian product

Definition

Let $I$ be a set and let $(\mathbf{A}_i)_{i \in I}$ be a family of algebras of the same signature, with universes $A_i$ . The direct product $\prod_{i \in I} \mathbf{A}_i$ has as universe the Cartesian product $\prod_{i \in I} A_i$ , 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.

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Tag	`68WXFBJEA6`
URI	`m2://68WXFBJEA6`
Kind	definition
Writing	W0 stub
MSC	08B25
Defines	Direct product of algebras
Uses	Algebraic structure, Cartesian product
Changed	2 November 2026

Direct product of algebras

Definition

Remarks

Reading path

Further reading

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