Submitted:
28 September 2018
Posted:
28 September 2018
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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.
Keywords:
conditional independence
; entropy function
; functional dependence
; lattice
; non-Shannon inequality
; polymatroid function
; subgroup
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