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Open AccessArticle

Entropy Inequalities for Lattices

Copenhagen Business College, Nørre Voldgade 34, 1358 Copenhagen K, Denmark
Current address: Rønne Alle 1, st., 2860 Søborg, Denmark.
Entropy 2018, 20(10), 784; https://doi.org/10.3390/e20100784
Received: 1 September 2018 / Revised: 26 September 2018 / Accepted: 10 October 2018 / Published: 12 October 2018
(This article belongs to the Special Issue Entropy and Information Inequalities)
We study entropy inequalities for variables that are related by functional dependencies. Although the powerset on four variables is the smallest Boolean lattice with non-Shannon inequalities, there exist lattices with many more variables where the Shannon inequalities are sufficient. We search for conditions that exclude the existence of non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group. In order to formulate and prove the results, one has to bridge lattice theory, group theory, the theory of functional dependences and the theory of conditional independence. It is demonstrated that the Shannon inequalities are sufficient for planar modular lattices. The proof applies a gluing technique that uses that if the Shannon inequalities are sufficient for the pieces, then they are also sufficient for the whole lattice. It is conjectured that the Shannon inequalities are sufficient if and only if the lattice does not contain a special lattice as a sub-semilattice. View Full-Text
Keywords: conditional independence; entropy function; functional dependence; lattice; non-Shannon inequality; polymatroid function; subgroup conditional independence; entropy function; functional dependence; lattice; non-Shannon inequality; polymatroid function; subgroup
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Harremoës, P. Entropy Inequalities for Lattices. Entropy 2018, 20, 784.

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