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  "body": "\n## Definition\n\nLet $A$ be a nonempty set.\nLet $\\Gamma$ be a set of finitary relations on $A$.\nA *primitive positive formula* over $\\Gamma$ is a formula of the form\n$$\\exists y_1 \\cdots \\exists y_k \\, \\bigl(\\varphi_1 \\wedge \\cdots \\wedge \\varphi_r\\bigr).$$\nHere each $\\varphi_j$ is an equation $z_1 = z_2$ or an atomic formula $R(z_1, \\dots, z_m)$ with $R \\in \\Gamma$ of arity $m$.\nThe $z_i$ are variables.\nA relation $S \\subseteq A^p$ is *primitive positive definable* from $\\Gamma$ if $S = \\set{(a_1, \\dots, a_p) \\in A^p : \\psi(a_1, \\dots, a_p)}$.\nHere $\\psi(x_1, \\dots, x_p)$ is a primitive positive formula over $\\Gamma$ with free variables $x_1, \\dots, x_p$.\nA *relational clone* on $A$ is a set $\\Gamma$ of finitary relations on $A$ such that:\n\n1. $\\Gamma$ contains the binary equality relation $\\set{(a, a) : a \\in A}$.\n2. $\\Gamma$ contains the empty relation of each arity.\n3. Every relation that is primitive positive definable from $\\Gamma$ belongs to $\\Gamma$.\n\n## Remarks\n\nExample: for any set $F$ of operations on $A$, the set $\\operatorname{Inv}(F)$ of all relations preserved by every $f \\in F$ is a relational clone.\nCheck: equality and empty relations are preserved by every operation.\nIf the relations in a primitive positive formula are preserved by $f$, so is the defined relation, because $f$ can be applied to the witnesses $y_i$ too.\nNon-example: on $\\set{0, 1}$, the set consisting of equality, the empty relations and the single relation $R = \\set{(0, 1)}$ is not a relational clone.\nThe formula $\\exists y \\, R(x, y)$ defines the unary relation $\\set{0}$, which is not in this set.\nConventions differ slightly between texts on whether the empty relation is included.\n",
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