 Razborov closure

Fix a finite ground set $V$, an integer $r\ge 2$ and a size limit $\ell$. The Razborov closure of a set family $\mathcal{F}\subseteq \{A\subseteq V:|A|\le \ell \}$ repeatedly adjoins a set $W$ of size at most $\ell$ whenever existing members $W_1,\dots ,W_r$ satisfyAll 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.^[ref]