= Razborov closure
{c}
Fix a finite ground <set> $V$, an <integer> $r\geq2$ and a size limit $\ell$. The <Razborov closure> of a <set family> $\mathcal F\subseteq\{A\subseteq V:|A|\leq\ell\}$ repeatedly adjoins a <set> $W$ of size at most $\ell$ whenever existing members $W_1,\ldots,W_r$ satisfy
$$
W_i\cap W_j\subseteq W\qquad(i\ne j).
$$
All adjoined <sets> remain in the same truncated ambient family. The process terminates because that family is finite. The final family is the least family containing $\mathcal F$ that is closed under the rule, so the result does not depend on the order of additions.
For $r=2$, the members $\{1,2\}$ and $\{1,3\}$ force $\{1\}$. In the <finite lattice approximation for monotone clique circuits>, this rule supplies a closure operation used to form approximate joins. It can add a smaller witness, which is why closure constrains the number of inclusion-minimal members.<Razborov's Method of Approximations by Timothy Gowers>[section 3.3 and section 4]{ref}
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