Paratopological groups (Q1605605)

A group \(G\) with topology \(\tau\) is called a paratopological group if the multiplication on the group \(G\) is continuous. The topology \(\tau\) in this case is called a paratopology. The author deals with properties of paratopological groups related, in particular, to cardinal invariants, metrization and minimality. Some of the main results of the paper are the following. Theorem on continuous epimorphism. Let \(G\) and \(H\) be paratopological groups, let \(\varphi\colon G\to H\) be a continuous epimorphism and let \(N=\text{Ker} \varphi\). Let a map \(\sigma\colon G/N\to H\) be defined as \(\sigma(xN)=\varphi(x)\). Then \(\sigma\) is a continuous isomorphism. Moreover, if the map \(\varphi\) is open, then \(\sigma\) is a topological isomorphism. Theorem on isomorphism. Let \(G\) be a paratopological group, let \(H\) be a subgroup of \(G\), and let \(N\) be a normal subgroup of \(G\). Then \(HN\) is a subgroup of \(G\), \(N\) is a normal subgroup of \(G\) and the map \(\sigma\colon H/(H\cap N)\to(NH)/N\) defined as \(\sigma(h(H\cap N))=hN\) is a homomorphic compression. Proposition 1.3 of the article implies that every Hausdorff SIN-paratopology (Small Invariant Neighborhoods) on a group can be weakened to a Hausdorff group SIN-topology. \textit{I. Guran} [Mat. Stud. 10, No. 2, 223-224 (1998; Zbl 0927.22007)] raised the question: can every Hausdorff paratopology on a group be weakened to a Hausdorff group topology? The author of the article under review proposes an example that gives a negative answer to this question. Moreover, he proposes the following proposition. Every Hausdorff ring SIN-topology on a quasifield can be weakened to a Hausdorff quasifield SIN-topology.