Lattices with non-Shannon Inequalities
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.
Cite
@article{arxiv.1502.04336,
title = {Lattices with non-Shannon Inequalities},
author = {Peter Harremoës},
journal= {arXiv preprint arXiv:1502.04336},
year = {2015}
}
Comments
Ten pages. Submitted to ISIT 2015. The appendix will not appear in the proceedings