# Hereditary Coreflective Subcategories in Certain Categories of Abelian Semitopological Groups

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Notation

## 3. Results

**Lemma**

**1.**

- 1.
- a finite cyclic group,
- 2.
- the discrete group of integers,
- 3.
- the indiscrete group of integers,
- 4.
- the group of integers with the topology generated by its subgroups of the form $\left(\right)$, where $n\in \mathbb{N}$, $p\in P$ and P is a given set of prime numbers.

**Proof.**

**Corollary**

**1.**

**Lemma**

**2.**

**Proof.**

**Example**

**1.**

**Lemma**

**3.**

**Proof.**

**Proposition**

**1.**

- 1.
- If every cyclic group from $\mathbf{A}$ of order ${p}_{i}^{{\alpha}_{i}}$ is homeomorphic to ${Z}_{{p}_{i}^{{\alpha}_{i}}}$ with the subspace topology induced from $r\left(\mathbb{Z}\right)$ or there exists a cyclic group ${Z}_{{p}_{i}^{{\beta}_{i}}}$ (where ${\beta}_{i}<{\alpha}_{i}$) from $\mathbf{A}$ such that $\overline{\left\{0\right\}}=\left(\right)open="\langle "\; close="\rangle ">{p}_{i}^{{m}_{i}}$ in ${Z}_{{p}_{i}^{{\beta}_{i}}}$, then let ${\mathbf{B}}_{i}$ be the subcategory consisting precisely of those groups from $\mathbf{A}$ that do not have an element of order ${p}_{i}^{{\alpha}_{i}}$.
- 2.
- If the subgroup ${Z}_{{p}_{i}^{{\alpha}_{i}}}$ of $r\left(\mathbb{Z}\right)$ is not indiscrete, let ${\mathbf{C}}_{i}$ be the subcategory consisting precisely of such groups G from $\mathbf{A}$ that if H is a cyclic subgroup of G of order ${p}_{i}^{{\beta}_{i}}$, where ${\beta}_{i}\le {\alpha}_{i}$, then the index of $\overline{\left\{{e}_{H}\right\}}$ in H is less than ${p}_{i}^{{m}_{i}}$.

**Proof.**

**Example**

**2.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

- 1.
- If the embedding $\u2329n\u232a\to r\left(\mathbb{Z}\right)$ is an $\mathbf{A}$-epimorphism, then the subcategory $\mathbf{B}$ of all torsion groups from $\mathbf{A}$ is the largest hereditary coreflective subcategory of $\mathbf{A}$ that is not bicoreflective in $\mathbf{A}$.
- 2.
- For every minimal natural number k such that $k|n$ and the embedding $\u2329k\u232a\to r\left(\mathbb{Z}\right)$ is not an $\mathbf{A}$-epimorphism let ${\mathbf{B}}_{k}$ be the subcategory consisting of such groups G from $\mathbf{A}$ that if H is a cyclic subgroup of G then the index of $\overline{\left\{{e}_{H}\right\}}$ in H is at most $\frac{n}{k}$. The subcategories ${\mathbf{B}}_{k}$ are maximal hereditary coreflective subcategories of $\mathbf{A}$ that are not bicoreflective in $\mathbf{A}$.Assume that for every minimal natural number k such that $k|n$ and the embedding $\u2329k\u232a\to r\left(\mathbb{Z}\right)$ is not an $\mathbf{A}$-epimorphism, $\mathbf{A}$ contains a finite cyclic group ${G}_{k}$ such that the index of $\overline{\left\{{e}_{{G}_{k}}\right\}}$ in ${G}_{k}$ is greater than $\frac{n}{k}$. Then the subcategory $\mathbf{B}$ of all torsion groups from $\mathbf{A}$ is also a maximal hereditary coreflective subcategory of $\mathbf{A}$ that is not bicoreflective in $\mathbf{A}$.

**Proof.**

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Pitrová, V.
Hereditary Coreflective Subcategories in Certain Categories of Abelian Semitopological Groups. *Axioms* **2019**, *8*, 85.
https://doi.org/10.3390/axioms8030085

**AMA Style**

Pitrová V.
Hereditary Coreflective Subcategories in Certain Categories of Abelian Semitopological Groups. *Axioms*. 2019; 8(3):85.
https://doi.org/10.3390/axioms8030085

**Chicago/Turabian Style**

Pitrová, Veronika.
2019. "Hereditary Coreflective Subcategories in Certain Categories of Abelian Semitopological Groups" *Axioms* 8, no. 3: 85.
https://doi.org/10.3390/axioms8030085