Irreducible Boolean Functions

TL;DR

This paper explores the structure of Boolean functions through a quasi-order called simple minor, identifying join-irreducible functions via hypergraph properties and linking them to Steiner systems and graphs.

Contribution

It introduces a combinatorial characterization of join-irreducible Boolean functions using hypergraph and Steiner system properties, expanding understanding of their structure.

Findings

Join-irreducibility corresponds to a hypergraph property called -2-monomorphism.

Steiner systems that are -2-monomorphic yield join-irreducible Boolean functions.

Specific graphs are characterized as representing join-irreducible members of the poset.

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 $\tilde{\Omega}$.