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  "body": "\n## Summary\n\nA relational clone on a set $A$ is a set of finitary relations on $A$ that is closed under primitive positive definitions.\nIt must also contain the equality relation and the empty relation.\nPrimitive positive definitions are those that use only conjunction, existential quantification and equality.\nRelational clones are dual to [[clones|PTKR703G6A]] of operations.\nOn a finite set, the operators $\\operatorname{Pol}$ and $\\operatorname{Inv}$ give a bijection between them.\nAnother common name is co-clone.\n\n## Explanations\n\n### L1 human\n\nA relational clone is a family of relations that is closed under the simplest ways of building new relations from old ones.\nOne may combine relations with \"and\", hide some coordinates with \"there exists\", and identify coordinates with \"equals\".\nFor example, from the order $\\le$ on a set one can define $\\set{(x, z) : \\exists y \\, (x \\le y \\wedge y \\le z)}$, and a relational clone containing $\\le$ contains this relation too.\nThe set of all relations preserved by a fixed set of operations is always a relational clone.\n",
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