# Supersymmetric Polynomials and a Ring of Multisets of a Banach Algebra

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Group Rings of Multisets

**Proposition**

**1.**

**Proof.**

**Definition**

**1.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Definition**

**2.**

**Proposition**

**4.**

**Definition**

**3.**

- 1.
- $\parallel z\parallel \ge 0$ and $\parallel z\parallel =0$ if and only if $z=0,$
- 2.
- $\parallel z+r\parallel \le \parallel z\parallel +\parallel r\parallel ,$
- 3.
- $\parallel -z\parallel =\parallel z\parallel ,$
- 4.
- $\parallel zr\parallel \le C\parallel z\parallel \parallel r\parallel $ for some constant $C>0.$

**Definition**

**4.**

**Proposition**

**5.**

**Proof.**

**Example**

**1.**

**Proposition**

**6.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 3. Homomorphisms and Supersymmetric Polynomials

**Theorem**

**2.**

**Proof.**

**Example**

**2.**

**Corollary**

**1.**

**Proof.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

**Theorem**

**3.**

**Proof.**

**Example**

**8.**

**Example**

**9.**

**Example**

**10.**

## 4. Discussions and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Chernega, I.; Zagorodnyuk, A.
Supersymmetric Polynomials and a Ring of Multisets of a Banach Algebra. *Axioms* **2022**, *11*, 511.
https://doi.org/10.3390/axioms11100511

**AMA Style**

Chernega I, Zagorodnyuk A.
Supersymmetric Polynomials and a Ring of Multisets of a Banach Algebra. *Axioms*. 2022; 11(10):511.
https://doi.org/10.3390/axioms11100511

**Chicago/Turabian Style**

Chernega, Iryna, and Andriy Zagorodnyuk.
2022. "Supersymmetric Polynomials and a Ring of Multisets of a Banach Algebra" *Axioms* 11, no. 10: 511.
https://doi.org/10.3390/axioms11100511