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  "body": "\n## Definition\n\nLet $I$ be a set and let $(\\mathbf{A}_i)_{i \\in I}$ be a family of [[algebras|QJGPT2W0FT]] of the same signature, with universes $A_i$.\nThe *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 $i$.\nEach $n$-ary fundamental operation $f$ is defined coordinatewise:\n$$f(a_1, \\dots, a_n)(i) = f^{\\mathbf{A}_i}(a_1(i), \\dots, a_n(i)) \\quad \\text{for all } i \\in I.$$\n\n## Remarks\n\nFor 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.\nFor two factors one writes $\\mathbf{A}_1 \\times \\mathbf{A}_2$, with elements written as pairs.\nThe product of the empty family is a one-element algebra.\n\nExample: the group $\\Z/2\\Z \\times \\Z/3\\Z$ has six elements and is cyclic, generated by $(1, 1)$.\nCoordinatewise, $k \\cdot (1, 1) = (k \\bmod 2, k \\bmod 3)$, which is $(0, 0)$ first for $k = 6$.\n\nNon-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.\nIn the product, $x + x = 0$ for every $x$, since this holds in each coordinate, while $1 + 1 = 2 \\ne 0$ in $\\Z/4\\Z$.\n\nA direct product of algebras is the product in the category of algebras of the given signature, by the universal property of the projections.\n",
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