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  "body": "\n## Definition\n\nLet $A$ be a nonempty set and let $\\mathcal{L}_A$ be the set of all [[clones|PTKR703G6A]] on $A$.\nThe *lattice of clones* on $A$ is the partially ordered set $(\\mathcal{L}_A, \\subseteq)$.\nFor a family $(C_i)_{i \\in I}$ of clones, its meet is $\\bigcap_{i \\in I} C_i$ and its join is the clone generated by $\\bigcup_{i \\in I} C_i$.\n\n## Remarks\n\nThe intersection of clones is a clone, since each clause of the definition of a clone is kept under intersections, so $\\mathcal{L}_A$ is a [[complete lattice|D7A0ZGZT5S]].\nExample: on $\\set{0, 1}$, the meet of the clones generated by $\\wedge$ and by $\\vee$ is the clone of [[projections|769B35T6WG]].\nTheir join is the clone of all monotone operations that preserve both $0$ and $1$.\nCheck for the meet: the members of the first clone are the meets $x_{i_1} \\wedge \\cdots \\wedge x_{i_k}$, and those of the second are joins.\nA meet of $k \\ge 2$ distinct variables differs from every join on some input, so only projections are in both.\nNon-example: the join is not the union in general.\nOn $\\set{0, 1}$, the union of the clones generated by $\\wedge$ and by $\\vee$ is not a clone, since it misses the majority operation $(x \\wedge y) \\vee (y \\wedge z) \\vee (x \\wedge z)$, which lies in their join.\nThe lattice of clones on any set is an algebraic lattice whose compact elements are the finitely generated clones.\n",
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