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 \emph{simple minor} quasi-order. We look at the join-irreducible members of the resulting poset $\tildeΩ$. 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 $\tildeΩ$ are the -2-monomorphic Steiner systems. We also describe the graphs which correspond to join-irreducible members of $\tildeΩ$.

The current manuscript constitutes an extension to the paper "Irreducible Boolean Functions" (arXiv:0801.2939v1)