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  "body": "\n## Summary\n\nThe lattice of clones on a set $A$ is the set of all [[clones|PTKR703G6A]] on $A$, ordered by inclusion.\nIt is a complete lattice: the meet of a family of clones is its intersection, and the join is the clone generated by the union.\nIts least element is the clone of [[projections|769B35T6WG]] and its greatest element is the clone of all finitary operations.\nOn a two-element set it is countable, and [[Post's lattice|MA20X7JCET]] describes it completely.\nOn a finite set with at least three elements it has the cardinality of the continuum.\n\n## Explanations\n\n### L1 human\n\nEach clone is a toolbox of operations, and a bigger clone can build more.\nOrdering the toolboxes by inclusion gives the lattice of clones.\nAt the bottom sits the clone that can only copy inputs; at the top sits the clone that can build every operation.\nOn $\\set{0, 1}$ there are countably many clones, while on $\\set{0, 1, 2}$ there are already uncountably many.\n",
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