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  "body": "\n## Definition\n\nLet $I$ be a nonempty set and let $F$ be a [[filter|15PF73BKY6]] on $I$.\nLet $(\\mathbf{A}_i)_{i \\in I}$ be a family of [[algebras|QJGPT2W0FT]] of the same signature, and let $\\mathbf{P} = \\prod_{i \\in I} \\mathbf{A}_i$ be their [[direct product|WJ73XYBNE4]].\nDefine $\\theta_F$ on the universe of $\\mathbf{P}$ by $(a, b) \\in \\theta_F$ if and only if $\\set{i \\in I : a(i) = b(i)} \\in F$.\nThe *reduced product* $\\prod_{i \\in I} \\mathbf{A}_i / F$ is the [[quotient algebra|N5KE7FH6CX]] $\\mathbf{P}/\\theta_F$.\n\n## Remarks\n\nThe relation $\\theta_F$ is a [[congruence|DCSXXT4SER]] on $\\mathbf{P}$.\nReflexivity uses $I \\in F$, and transitivity uses that $F$ is closed under finite intersections and supersets.\nCompatibility holds because $\\set{i : f(a_1, \\dots, a_n)(i) = f(b_1, \\dots, b_n)(i)}$ contains the intersection of the sets $\\set{i : a_k(i) = b_k(i)}$.\n\nExample: if $F = \\set{I}$, then $\\theta_F$ is equality and the reduced product is the direct product itself.\nIf $F = \\set{X \\subseteq I : J \\subseteq X}$ for some nonempty $J \\subseteq I$, the reduced product is isomorphic to $\\prod_{j \\in J} \\mathbf{A}_j$.\n\nNon-example: on $I = \\set{1, 2, 3}$, the family $G = \\set{\\set{1, 2}, \\set{2, 3}, I}$ is not a filter, since $\\set{1, 2} \\cap \\set{2, 3} = \\set{2} \\notin G$.\nWith it, the analogous relation on $\\Z^3$ is not transitive: $(0, 0, 0)$ is related to $(0, 0, 1)$, which is related to $(1, 0, 1)$, but $(0, 0, 0)$ and $(1, 0, 1)$ agree only on $\\set{2}$.\n\nWhen $F$ is an ultrafilter, the reduced product is the [[ultraproduct|AX9Q86GB8M]] $\\prod_{i \\in I} \\mathbf{A}_i / F$.\n",
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