# Entropy Inequalities for Lattices

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Lattices of Functional Dependence

**Example**

**1.**

**Proposition**

**1.**

**Example**

**2.**

**Proposition**

**2.**

**Remark**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

## 3. Polymatroid Functions and Separoids

**Definition**

**1.**

**Example**

**3.**

**Proposition**

**3.**

**Proof.**

**Remark**

**2.**

**Proposition**

**4.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Proposition**

**5.**

**Proposition**

**6.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**2.**

**Proof.**

## 4. Entropy in Functional Dependence Lattices

**Definition**

**2.**

**Definition**

**3.**

**Example**

**4.**

**Proposition**

**7.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**2**, or by ${M}_{5}$, or by ${M}_{6}$, or by ${M}_{7}$. All these lattices are representable, and thereby, they are Shannon lattices. □

**Conjecture**

**1.**

## 5. The Skeleton of a Lattice

**Definition**

**4.**

**Lemma**

**2.**

**Proof.**

**Proposition**

**8.**

## 6. Results for Planar Lattices

**Theorem**

**9.**

**Proof.**

**2**as the factor lattice. These two blocks are glued together along a chain ${L}_{j}={y}_{1}\subset {y}_{2}\subset \cdots \subset {y}_{t}={R}_{k}$ that ${\mathcal{L}}_{1}$ and ${\mathcal{L}}_{0}$ share. There are two cases: either ${R}_{k}\subset {R}_{n}$ or ${R}_{k}={R}_{n}.$

**Theorem**

**10.**

**Proof.**

**2**, or to the lattice $\mathbf{2}\times \mathbf{2}$, or to one of the lattices ${M}_{n}.$ □

**2**or to the lattice $\mathbf{2}\times \mathbf{2}.$ Therefore, the lattice is a sublattice of a product of two chains, as illustrated in Figure 5. This result was first proven by Dilworth [38]. Other characterizations of planar distributive lattices can be found in the literature [39]. Since the extreme polymatroid functions on the lattices

**2**and the lattice $\mathbf{2}\times \mathbf{2}$ only take the values zero and one, the same is true for any planar distributive lattice.

## 7. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Augmentation

## Appendix B. Lattices of Size 1–7

#### Appendix B.1. Lattice of Size 1

#### Appendix B.2. Lattice of Size 2

**2**is the only lattice of Size 2.

#### Appendix B.3. Lattices of Size 3

**3**is only one lattice of Size 3, and and it is distributive. The extreme polymatroid functions can be represented by the lattice

**2**.

#### Appendix B.4. Lattices of Size 4

**2**.

#### Appendix B.5. Lattices of Size 5

**2**.

#### Appendix B.6. Lattices of Size 6

**2**.

**2**. The first two lattices are modular, but not distributive. The next three are not modular.

**2**. The first four are not modular.

#### Appendix B.7. Lattices of Size 7

**2**.

**2**. The first two lattices are modular. The last five lattices are not modular.

**2**. The first five lattices are modular.

## References

- Zhang, Z.; Yeung, R.W. On characterization of entropy function via information inequalities. IEEE Trans. Inform. Theory
**1998**, 44, 1440–1452. [Google Scholar] [CrossRef] - Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423, 623–656. [Google Scholar] [CrossRef] - McGill, W. Multivariate information transmission. Psychometrika
**1954**, 19, 97–116. [Google Scholar] [CrossRef] - Yeung, R.W. A First Course in Information Theory; Kluwer: New York, NY, USA, 2002. [Google Scholar]
- Stern, M. Semimodular Lattices. Theory and Applications; Number 73 in Encyclopedia of Mathematics and Its Applications; Cambridge University Press: Cambridge, UK, 1999. [Google Scholar]
- Chan, T.H.; Yeung, R.W. On a Relation between Information Inequalities and Group Theory. IEEE Trans. Inform. Theory
**2002**, 48, 1992–1995. [Google Scholar] [CrossRef] - Harremoës, P. Functional Dependences and Bayesian Networks. In Proceedings of the WITMSE 2011; Number Report C-2011-45 in Series of Publications C; Rissanen, J., Myllymäki, P., Teemu Roos, I.T., Yamanishi, K., Eds.; Department Computer Science, University of Helsinki: Helsinki, Finland, 2011; pp. 35–38. [Google Scholar]
- Harremoës, P. Lattices with non-Shannon inequalities. In Proceedings of the 2015 IEEE International Symposium on Information Theory, Hong Kong, China, 14–19 June 2015; pp. 740–744. [Google Scholar] [CrossRef]
- Lee, T.T. An algebraic theory of relational databases. Bell Syst. Tech. J.
**1983**, 62, 3159–3204. [Google Scholar] [CrossRef] - Demetrovics, J.; Libkin, L.; Muchnik, I.B. Functional dependencies and the semilattice of closed classes. In Proceedings of the 2nd Symposium on Mathematical Fundamentals of Database Systems (MFDBS ’89), Visegrád, Hungary, 26–30 June 1989; Springer: Berlin, Germany, 1989; pp. 136–147. [Google Scholar]
- Matúš, F. Abstract functional dependency structures. Theor. Comput. Sci.
**1991**, 81, 117–126. [Google Scholar] [CrossRef] - Demetrovics, J.; Libkin, L.; Muchnik, I.B. Functional Dependencies in Relational Databases: A Lattice Point of View. Discret. Appl. Math.
**1992**, 40, 155–185. [Google Scholar] [CrossRef] - Levene, M. A Lattice View of Functional Dependencies in Incomplete Relations. Acta Cybern.
**1995**, 12, 181–207. [Google Scholar] - Thakor, S.; Chan, T.; Grant, A. A minimal set of Shannon-type inequalities for functional dependence structures. In Proceedings of the 2017 IEEE International Symposium on Information Theory (ISIT), Aachen, Germany, 25–30 June 2017; pp. 679–683. [Google Scholar] [CrossRef]
- Chan, T.; Thakor, S.; Grant, A. A Minimal Set of Shannon-type Inequalities for MRF Structures with Functional Dependencies. In Proceedings of the 2018 IEEE International Symposium on Information Theory (ISIT), Vail, CO, USA, 17–22 June 2018; pp. 1759–1763. [Google Scholar] [CrossRef]
- Harremoës, P. Time and Conditional Independence; IMFUFA-Tekst, IMFUFA Roskilde University. 1993, Volume 255. Original in Danish Entitled Tid og Betinget Uafhængighed. An English Translation of Some of the Chapters. Available online: http://www.harremoes.dk/Peter/afh/afhandling.pdf (accessed on 10 October 2018).
- Caspard, N.; Monjardet, B. The lattices of closure systems, closure operators, and implicational systems on a finite set: A survey. Discret. Appl. Math.
**2003**, 127, 241–269. [Google Scholar] [CrossRef] - Grätzer, G. General Lattice Theory, 2nd ed.; Birkhäuser: Basel, Switzerland, 2003. [Google Scholar]
- Armstrong, W.W. Dependency Structures of Data Base Relationships. In Proceedings of the IFIP Congress, Stockholm, Sweden, 5–10 August 1974; pp. 580–583. [Google Scholar]
- Ullman, J.D. Principles of Database and Knowledge-Base Systems; Computer Science Press: Stanford, CA, USA, 1989; Volume 1. [Google Scholar]
- Levene, M.; Loizou, G. A Guide Tour of Relational Databases and Beyond; Springer: Berlin, Germany, 1999. [Google Scholar]
- Whitman, P.M. Lattices, equivalence relations, and subgroups. Bull. Am. Math. Soc.
**1946**, 52, 507–522. [Google Scholar] [CrossRef] - Shannon, C. The lattice theory of information. Trans. IRE Prof. Group Inf. Theory
**1953**, 1, 105–107. [Google Scholar] [CrossRef] - Dawid, A.P. Separoids: A mathematical framework for conditional independence and irrelevance. Ann. Math. Artif. Intell.
**2001**, 32, 335–372. [Google Scholar] [CrossRef] - Constantinou, P.; Dawid, A.P. Extended Conditional Independence and Applications in Causal Inference. Ann. Stat.
**2017**, 45, 1–36. [Google Scholar] [CrossRef] - Paolini, G. Independence Logic and Abstract Independence Relations. Math. Logic Q.
**2015**, 61, 202–216. [Google Scholar] [CrossRef] - Pearl, J. Probabilistic Reasoning in Intelligent Systems; Morgan Kaufmann Publ.: San Mateo, CA, USA, 1988. [Google Scholar]
- Studený, M. Probabilistic Conditional Independence Structures; Springer: Belin, Germany, 2005. [Google Scholar]
- Studený, M. Conditional Independence Relations Have No Finite Complete Characterization. 1990. Available online: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.51.7014&rep=rep1&type=pdf (accessed on 10 October 2018).
- Schmidt, R. Subgroup Lattices of Groups; Walter de Gruyter: Berlin, Germany, 1994. [Google Scholar]
- Matúš, F. Infinitely many information inequalities. In Proceedings of the 2007 IEEE International Symposium on Information Theory, Nice, France, 24–29 June 2007; pp. 2101–2105. [Google Scholar] [CrossRef]
- Vámos, P. The Missing Axiom of Matroid Theory is Lost Forever. J. Lond. Math. Soc.
**1978**, 18, 403–408. [Google Scholar] [CrossRef] - Mayhew, D.; Whittle, G.; Newman, M. Is the Missing Axiom of Matroid Theory Lost Forever? Q. J. Math.
**2014**, 65, 1397–1415. [Google Scholar] [CrossRef] - Mayhew, D.; Newman, M.; Whittle, G. Yes, the “missing axiom” of matroid theory is lost forever. Trans. Am. Math. Soc.
**2018**, 370, 5907–5929. [Google Scholar] [CrossRef][Green Version] - Czédli, G. Factor lattices by tolerance. Acta Sci. Math.
**1982**, 44, 35–42. [Google Scholar] - Hermann, C. S-verklebte Summen von Verbänden. Math. Z.
**1973**, 130, 255–274. [Google Scholar] [CrossRef] - Quackenbush, R.W. Planar Lattices. In Proceedings of the University of Houston Lattice Theory Conference 1973, Houston, TX, USA, 22–24 March 1973. [Google Scholar]
- Dilworth, R.P. A decomposition theorem for partially ordered sets. Ann. Math.
**1950**, 51, 161–166. [Google Scholar] [CrossRef] - Chen, C.C.; Koh, K.M. A characterization of finite distributive planar lattices. Discret. Math.
**1973**, 5, 207–213. [Google Scholar] [CrossRef] - Quackenbush, G.G.W. The variety generated by planar modular lattices. Algebra Universalis
**2010**, 63, 187–201. [Google Scholar] [CrossRef] - Guille, L.; Chan, T.; Grant, A. The Minimal Set of Ingleton Inequalities. IEEE Trans. Inform. Theory
**2011**, 57, 1849–1864. [Google Scholar] [CrossRef][Green Version] - Paajanen, P. Finite p-Groups, Entropy Vectors, and the Ingleton Inequality for Nilpotent Groups. IEEE Trans. Inf. Theory
**2014**, 60, 3821–3824. [Google Scholar] [CrossRef] - Harremoës, P. Influence Diagrams as Convex Geometries. Available online: http://www.harremoes.dk/Peter/FunctionalDAG.pdf (accessed on 21 September 2018).

**Figure 2.**The Matúš lattice with a non-entropic polymatroid function. This lattice is named in honor of František Matúš, who passed away shortly before the submission of this manuscript.

**Figure 4.**The skeleton of the lattice in the previous figure. It consist of four blocks glued together by the factor lattice illustrated to the right.

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Harremoës, P. Entropy Inequalities for Lattices. *Entropy* **2018**, *20*, 784.
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Harremoës P. Entropy Inequalities for Lattices. *Entropy*. 2018; 20(10):784.
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**Chicago/Turabian Style**

Harremoës, Peter. 2018. "Entropy Inequalities for Lattices" *Entropy* 20, no. 10: 784.
https://doi.org/10.3390/e20100784