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  "body": "\n## Summary\n\nThe direct product of a family of [[algebras|QJGPT2W0FT]] of the same signature is the [[Cartesian product|09CEDJYBKP]] of their universes with the operations computed coordinatewise.\nIt generalizes the [[direct product of groups|00HF4RJ5QN]], of rings and of vector spaces.\nProducts are one of the three basic constructions of universal algebra, with subalgebras and homomorphic images, and varieties are the classes closed under all three.\n\n## Explanations\n\n### L1 human\n\nTo multiply two algebras, take pairs, one element from each, and do every operation separately in each coordinate.\nFor example, in $\\Z/2\\Z \\times \\Z/3\\Z$ one adds $(1, 2) + (1, 2) = (0, 1)$, computing modulo $2$ in the first place and modulo $3$ in the second.\nAny identity that holds in every factor, such as commutativity, still holds in the product.\nProperties that are not identities can fail: a product of two fields is not a field, since $(1, 0) \\cdot (0, 1) = (0, 0)$.\n",
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