# 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.

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**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.
https://doi.org/10.3390/e20100784

**AMA Style**

Harremoës P.
Entropy Inequalities for Lattices. *Entropy*. 2018; 20(10):784.
https://doi.org/10.3390/e20100784

**Chicago/Turabian Style**

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