Join-Irreducible Boolean Functions

Abstract

This paper is a contribution to the study of a quasi-order on the set Ω of Boolean functions, the 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 \(\tilde{\Omega}\).