Abstract
This paper is a contribution to the study of a quasi-order on the set Omega of Boolean functions, the simple minor quasi-order. We look at the join-irreducible members of the resulting poset (Omega) over tilde. 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 (Omega) over tilde are the -2-monomorphic Steiner systems. We also describe the graphs which correspond to join-irreducible members of (Omega) over tilde.