Preprint Article Version 1 This version is not peer-reviewed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.

Journal reference: Entropy 2018, 20, 784
DOI: 10.3390/e20100784

Abstract

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.

Subject Areas

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

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