  This paper is a contribution to the study of a quasi-order on the set
$\Omega$ of Boolean functions, the \emph{simple minor} quasi-order. We look at
the join-irreducible members of the resulting poset $\tilde{\Omega}$. Using a
two-way correspondence between Boolean functions and hypergraphs,
join-irreducibility translates into a combinatorial property of hypergraphs. We
observe that among Steiner systems, those which yield join-irreducible members
of $\tilde{\Omega}$ are the -2-monomorphic Steiner systems. We also describe
the graphs which correspond to join-irreducible members of $\tilde{\Omega}$.
