Irreducible Boolean Functions

Abstract

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 $\tilde{\Omega }$.

10 pages, ROGICS08,Mahdia 12-15 may 2008