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  "body": "\n## Definition\n\nLet $A$ be a nonempty set.\nFor $n \\ge 1$, let $O_A^{(n)}$ be the set of all $n$-ary [[operations|EJ8YJVM9BW]] $A^n \\to A$, and let $O_A = \\bigcup_{n \\ge 1} O_A^{(n)}$.\nA *clone* on $A$ is a subset $C \\subseteq O_A$ such that:\n\n1. $C$ contains every [[projection|769B35T6WG]] $\\pi^n_k$ with $1 \\le k \\le n$.\n2. $C$ is closed under [[composition|0THBK9V825]]: if $f \\in C$ is $m$-ary and $g_1, \\dots, g_m \\in C$ are $n$-ary, then the $n$-ary operation\n$$h(a_1, \\dots, a_n) = f\\bigl(g_1(a_1, \\dots, a_n), \\dots, g_m(a_1, \\dots, a_n)\\bigr)$$\nis in $C$.\n\n## Remarks\n\nExample: the set of all operations $f(x_1, \\dots, x_n) = c_1 x_1 + \\cdots + c_n x_n$ on $\\Z$ with integer coefficients $c_i$ is a clone.\nCheck: $\\pi^n_k$ has $c_k = 1$ and the other $c_i = 0$, and substituting linear forms into a linear form gives a linear form.\nNon-example: the set of all operations $f(x_1, \\dots, x_n) = x_1 + \\cdots + x_n$ on $\\Z$, one for each $n \\ge 1$, together with all projections, is not a clone.\nIt contains $f(x_1, x_2) = x_1 + x_2$, so a composition $f(f(\\pi^2_1, \\pi^2_2), \\pi^2_1)$ gives $2x_1 + x_2$, which is not in the set.\nThe set of all [[term operations|N9M2N27END]] of an algebra $(A, F)$ is the smallest clone containing $F$, the clone generated by $F$.\nSome authors also allow nullary operations in clones.\n",
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