Join-irreducible Boolean functions

Details Join-irreducible Boolean functions Join-irreducible Boolean functions

2009-03-23

arxiv.org/abs/0903.3848v1

Mathematics

Combinatorics

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

Scientific paper

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 $\tilde{\Omega}$.

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