Equivalence of some factorization properties in topological algebra

Abstract

We show that the original concept of $\mathbb{R}$-factorizability, as well as some of its modifications, examined in the realms of topological, paratopological, and semitopological groups possess an essential feature of \emph{absoluteness} when transitioning to a broader category. This resolves a certain ambiguity in the research conducted to date and enables us to keep \lq{old\rq} notation for formally different notions of factorizability.

It is also shown that a paratopological group $G$ is $\mathbb{R}$-factorizable if and only if its $T_i$-reflection, $T_i(G)$, is $\mathbb{R}$-factorizable for some (equivalently, for each) $i\in{0,1,2,3\}$, which in turn is equivalent to the regular reflection $Reg(G)$ of $G$ being $\mathbb{R}$-factorizable. When substituting the aforementioned reflections with the quotient group $G/N$, where $N$ is the closure of the singleton $\{e_G\}$, this result holds true for every topological group $G$. The latter results indicate a specific form of stability regarding the concept of $\mathbb{R}$-factorizability.