# Schützenberger Symmetries in Network Structures

## Abstract

**:**

## 1. Introduction

## 2. Semigroups

## 3. The Semigroup ${\mathit{N}}_{\mathit{V}}$

**Lemma**

**1.**

**Lemma**

**2.**

## 4. Green’s Equivalence Classifications of ${\mathit{N}}_{\mathit{V}}$ and Associated Invariants

**Lemma**

**3.**

**Theorem**

**1.**

- i.
- $\mathrm{L}\left(E\right)=\mathrm{L}\left(F\right)$if and only if${O}_{r}\left(E\right)={O}_{r}\left(F\right);$
- ii.
- $\mathrm{R}\left(E\right)=\mathrm{R}\left(F\right)$if and only if${I}_{c}\left(E\right)={I}_{c}\left(F\right);$
- iii.
- $\mathrm{H}\left(\mathrm{E}\right)=\mathrm{H}\left(\mathrm{F}\right)$if and only if${O}_{r}\left(E\right)={O}_{r}\left(F\right)$and${I}_{c}\left(E\right)={I}_{c}\left(F\right)$;
- iv.
- $\mathrm{D}\left(\mathrm{E}\right)=\mathrm{D}\left(\mathrm{F}\right)$if and only if$\left(P\left(E\right),\subseteq \right)$and$\left(P\left(F\right),\subseteq \right)$are lattice isomorphic.

## 5. The Schützenberger Symmetries and Groups for an $\mathit{\U0001d4d7}$-Equivalence Class of Networks

**Theorem**

**2.**

**Theorem**

**3.**

## 6. Example: Solving the Right Schützenberger Problem for Simple Two Node Networks

## 7. Example: Solving the Schützenberger Problems for an $\mathit{\U0001d4d7}$-Class of Map Networks

## 8. Concluding Remarks

## Funding

## Acknowledgments

## Conflicts of Interest

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Parks, A.D.
Schützenberger Symmetries in Network Structures. *Symmetry* **2018**, *10*, 260.
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Parks AD.
Schützenberger Symmetries in Network Structures. *Symmetry*. 2018; 10(7):260.
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2018. "Schützenberger Symmetries in Network Structures" *Symmetry* 10, no. 7: 260.
https://doi.org/10.3390/sym10070260