Bottleneck extrema

Let E be a finite set. Call a family of mutually noncomparable subsets of E a clutter on E. It is shown that for any clutter $ℛ$ on E, there exists a unique clutter $ℒ$ on E such that, for any function f from E to real numbers,

$\begin{array}{c}minmaxf\left(x\right)=\\ R{\in }^{ℛ}x{\in }^R\end{array}\begin{array}{c}maxminf\left(x\right)\\ S{\in }^{ℒ}x{\in }^S\end{array}$

Specifically, $ℒ$ consists of the minimal subsets of E that have non-empty intersection with every member of $ℛ$. The pair $\left(ℛ,ℒ\right)$ is called a blocking system on E. An algorithm is described and several examples of blockings systems are discussed.