# Factoring Continuous Homomorphisms Defined on Submonoids of Products of Topologized Monoids

## Abstract

**:**

## 1. Introduction

**Proposition**

**1.**

#### Notation and Preliminary Facts

**Lemma**

**1.**

**Lemma**

**2.**

**Definition**

**1.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

## 2. Factoring Continuous Homomorphisms

**Theorem**

**1.**

**Proof.**

**Claim**

**1.**

**Claim**

**2.**

**Claim**

**3.**

**Claim**

**4.**

**Corollary**

**1.**

**Lemma**

**5.**

**Proof.**

**Claim**

**5.**

**Lemma**

**6.**

**Lemma**

**7.**

**Proposition**

**2.**

- (a)
- $\mathsf{\Sigma}D\subset S$, or
- (b)
- S is ω-retractable.

**Proof.**

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Proposition**

**3.**

**Lemma**

**8.**

**Lemma**

**9.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Example**

**1.**

**Problem**

**1.**

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Tkachenko, M.
Factoring Continuous Homomorphisms Defined on Submonoids of Products of Topologized Monoids. *Axioms* **2019**, *8*, 86.
https://doi.org/10.3390/axioms8030086

**AMA Style**

Tkachenko M.
Factoring Continuous Homomorphisms Defined on Submonoids of Products of Topologized Monoids. *Axioms*. 2019; 8(3):86.
https://doi.org/10.3390/axioms8030086

**Chicago/Turabian Style**

Tkachenko, Mikhail.
2019. "Factoring Continuous Homomorphisms Defined on Submonoids of Products of Topologized Monoids" *Axioms* 8, no. 3: 86.
https://doi.org/10.3390/axioms8030086