Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Entropy Inequalities for Lattices

Version 1 : Received: 1 September 2018 / Approved: 3 September 2018 / Online: 3 September 2018 (13:42:08 CEST)
Version 2 : Received: 28 September 2018 / Approved: 28 September 2018 / Online: 28 September 2018 (12:56:55 CEST)

A peer-reviewed article of this Preprint also exists.

Harremoës, P. Entropy Inequalities for Lattices. Entropy 2018, 20, 784. Harremoës, P. Entropy Inequalities for Lattices. Entropy 2018, 20, 784.


We study the existence or absence of non-Shannon inequalities for variables that are related by functional dependencies. Although the power-set on four variables is the smallest Boolean lattice with non-Shannon inequalities there exist lattices with many more variables without non-Shannon inequalities. We search for conditions that excludes the existence of non-Shannon inequalities. It is demonstrated that planar modular lattices cannot have 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.


conditional independence; entropy function; functional dependence; lattice; non-Shannon inequality; polymatroid function; subgroup


Computer Science and Mathematics, Probability and Statistics

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