Equivalence of some factorization properties in topological algebra

We show that the original concept of R-factorizability, as well as some of its modifications, examined in the realms of topological, paratopological, and semitopological groups possess an essential feature of absoluteness when transitioning to a broader category. This resolves a certain ambiguity in the research conducted to date and enables us to keep ‘old’ notation for formally different notions of factorizability. It is also shown that a paratopological group G is R-factorizable if and only if itis T-reflectioin, T (G), i s R-factorizable for some (equivalently, for each) i ∈ {0, 1, 2, 3}, which in turn is equivalent to the regular reflection Reg(G) of G being R-factorizable. When substituting the aforementioned reflections with the quotient group G/N, where N is the closure of the singleGton {e}, this result holds true for every topological group G. The latter results indicate a specific form of stability regarding the concept of R-factorizability. Several routes for further investigation are outlined at the end of the article.