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  "body": "\n## Definition\n\nLet $A$ be a set with at least two elements, and let $J_A$ be the clone of all [[projections|769B35T6WG]] on $A$.\nA [[clone|PTKR703G6A]] $C$ on $A$ is *minimal* if $C \\ne J_A$ and the only clones $D$ with $J_A \\subseteq D \\subseteq C$ are $J_A$ and $C$.\nEquivalently, $C$ is an atom of the [[lattice of clones|EX9NCDFYC7]] on $A$.\n\n## Remarks\n\nA clone $C \\ne J_A$ is minimal exactly when every non-projection operation in $C$ generates $C$.\nExample: on $\\set{0, 1}$, the clone generated by $\\wedge$ is minimal.\nCheck: its $n$-ary members are the operations $x_{i_1} \\wedge \\cdots \\wedge x_{i_r}$, and identifying variables in one with $r \\ge 2$ distinct variables gives $x \\wedge y$ back.\nNon-example: the clone generated by $\\wedge$ and $\\vee$ on $\\set{0, 1}$ is not minimal.\nIt properly contains the clone generated by $\\wedge$, which is not the projection clone.\nOn $\\set{0, 1}$ there are exactly seven minimal clones, generated by the constant $0$, the constant $1$, $\\neg$, $\\wedge$, $\\vee$, the majority operation, and $x + y + z$ modulo $2$.\n",
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