Entropy Inequalities for Lattices.

Area of Science:

• Information Theory
• Lattice Theory
• Group Theory
• Algebraic Statistics

Background:

• Shannon inequalities are fundamental in information theory for bounding entropy.
• Non-Shannon inequalities exist in some complex variable dependency structures.
• Understanding these inequalities is crucial for information processing and causal inference.

Purpose of the Study:

• To identify conditions under which Shannon inequalities are sufficient for functional dependencies.
• To investigate the relationship between non-Shannon inequalities and lattice structures.
• To explore connections between lattice theory, group theory, and conditional independence.

Main Methods:

• Analysis of Boolean lattices and their properties.
• Application of group theory to lattice structures.
• Development of a gluing technique for proving sufficiency of inequalities.
• Investigation of functional dependencies and conditional independence.

Main Results:

• Shannon inequalities are proven sufficient for planar modular lattices.
• A gluing technique demonstrates sufficiency for composite lattices.
• Conditions for the absence of non-Shannon inequalities are explored.
• A conjecture is proposed relating Shannon inequality sufficiency to lattice structure.

Conclusions:

• Planar modular lattices satisfy Shannon inequalities.
• The study bridges multiple mathematical fields to address information-theoretic problems.
• Further research is needed to confirm the conjecture on lattice sub-semilattices.