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Join-Irreducible Boolean Functions

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Abstract

This paper is a contribution to the study of a quasi-order on the set Ω of Boolean functions, the 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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Correspondence to Maurice Pouzet.

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The research of the first and last author has been supported by CMCU Franco-Tunisien “Outils mathématiques pour l’informatique”.

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Bouaziz, M., Couceiro, M. & Pouzet, M. Join-Irreducible Boolean Functions. Order 27, 261–282 (2010). https://doi.org/10.1007/s11083-010-9175-z

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