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  "body": "\n## Definition\n\nLet $A$ be an [[algebra|QJGPT2W0FT]] in which every operation has arity at least $1$.\nA *fuzzy subset* of $A$ is a [[function|0EPAT51DXQ]] $\\mu \\colon A \\to [0, 1]$; the number $\\mu(a)$ is the degree of membership of $a$.\nA fuzzy subset $\\mu$ is a *fuzzy [[subalgebra|7CHG6SD365]]* of $A$ if\n$$\\mu(f(a_1, \\dots, a_n)) \\ge \\min\\set{\\mu(a_1), \\dots, \\mu(a_n)}$$\nfor every operation $f$ of $A$ of arity $n$ and all $a_1, \\dots, a_n \\in A$.\nFor a [[group|G5S1M9]] $G$ with the operations $(x, y) \\mapsto xy$ and $x \\mapsto x^{-1}$, a fuzzy subalgebra is a *fuzzy subgroup*: $\\mu(xy) \\ge \\min\\set{\\mu(x), \\mu(y)}$ and $\\mu(x^{-1}) \\ge \\mu(x)$ for all $x, y \\in G$.\n\n## Remarks\n\nA fuzzy subset $\\mu$ is a fuzzy subalgebra if and only if each level set $\\mu_t = \\set{a \\in A : \\mu(a) \\ge t}$, for $t \\in [0, 1]$, is empty or a subalgebra.\nIndeed, if $a_1, \\dots, a_n \\in \\mu_t$ then $\\min_i \\mu(a_i) \\ge t$; conversely, apply the condition on $\\mu_t$ with $t = \\min_i \\mu(a_i)$.\nA fuzzy subset that takes only the values $0$ and $1$ is a fuzzy subalgebra exactly when $\\set{a : \\mu(a) = 1}$ is empty or a subalgebra.\n\nExample: on the group $(\\Z, +)$ let $\\mu(n) = 1$ for even $n$ and $\\mu(n) = 1/2$ for odd $n$.\nIf $m$ or $n$ is odd, the minimum is $1/2$ and $\\mu(m + n) \\ge 1/2$ holds; if both are even, $m + n$ is even and $\\mu(m + n) = 1$.\nAlso $-n$ has the parity of $n$, so $\\mu(-n) = \\mu(n)$.\nSo $\\mu$ is a fuzzy subgroup.\n\nNon-example: on $(\\Z, +)$ let $\\mu(1) = 1$ and $\\mu(n) = 0$ for $n \\ne 1$.\nThen $\\mu(1 + 1) = 0 \u003c 1 = \\min\\set{\\mu(1), \\mu(1)}$, so $\\mu$ is not a fuzzy subgroup.\n",
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