Lattices with non-Shannon Inequalities

TL;DR

This paper investigates the conditions under which non-Shannon inequalities exist in variable lattices, showing certain lattice structures cannot have such inequalities and exploring their relation to subgroup lattices.

Contribution

It identifies specific lattice classes that cannot have non-Shannon inequalities and links the existence of these inequalities to subgroup lattice isomorphisms.

Findings

3D distributive lattices lack non-Shannon inequalities

Planar modular lattices lack non-Shannon inequalities

Existence of non-Shannon inequalities relates to subgroup lattice isomorphism

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 ensures that no non-Shannon inequalities exist. It is demonstrated that 3-dimensional distributive lattices cannot have non-Shannon inequalities and 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.