Epireflections in topological algebraic structures

TL;DR

This paper investigates how epireflective functors behave within topological algebraic structures, establishing conditions for preservation of properties and solving an open question about epireflections.

Contribution

It provides new criteria for when epireflective functors preserve algebraic structures and properties, and resolves an open problem regarding epireflection coincidence.

Findings

Epireflective functors always preserve semi-topological structures.

Sufficient conditions are identified for topological algebraic structures.

The open question about the coincidence of epireflections is solved.

Abstract

Let $\sR$ be an epireflective category of $\topo$ and let $F_\sR$\, be the epireflective functor associated with $\sR$. If $\sA$ denotes a (semi)topological algebraic subcategory of $\topo$, we study when $F_\sR\,(A)$ is an epireflective subcategory of $\sA$. We prove that this is always the case for semi-topological structures and we find some sufficient conditions for topological algebraic structures. We also study when the epireflective functor preserves products, subspaces and other properties. In particular, we solve an open question about the coincidence of epireflections proposed by Echi and Lazar in \cite[Question 1.6]{Echi:MPRIA} and repeated in \cite[Question 1.9]{Echi:TP}. Finally, we apply our results in different specific topological algebraic structures.