Join-irreducible Boolean functions

TL;DR

This paper characterizes join-irreducible Boolean functions within a quasi-order framework, using hypergraph and Steiner system properties to identify key combinatorial structures.

Contribution

It introduces a novel combinatorial characterization of join-irreducible Boolean functions via hypergraph and Steiner system analysis.

Findings

Join-irreducibility corresponds to a specific hypergraph property.

Certain Steiner systems, called -2-monomorphic, yield join-irreducible functions.

The paper describes graphs associated with these join-irreducible 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 $\tilde{\Omega}$.