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  "body": "\n## Summary\n\nA reduced product of [[algebras|QJGPT2W0FT]] is a quotient of their [[direct product|WJ73XYBNE4]].\nIt identifies two elements when they agree on a \"large\" set of indices.\nA [[filter|15PF73BKY6]] on the index set says which sets count as large.\nWhen the filter is an [[ultrafilter|J64Y3JC0RE]], the reduced product is an [[ultraproduct|AX9Q86GB8M]].\nReduced products preserve all identities and, more generally, all Horn sentences, which makes them useful in universal algebra and model theory.\n\n## Explanations\n\n### L1 human\n\nElements of a product are lists with one entry from each factor.\nIn a reduced product, two lists count as equal if they agree almost everywhere, where \"almost everywhere\" is fixed in advance by a filter.\nFor example, with the filter of cofinite subsets of $\\N$, two sequences are equal when they differ in only finitely many places.\nThe operations are still computed entry by entry.\n",
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