Transversal, $T_{1}$-independent, and $T_{1}$-complementary paratopological group topologies

We discuss the class of paratopological groups which admits a transversal, $T_{1}$-independent and $T_{1}$-complementary paratopological group topology. We show that the Sorgenfrey line does not admit a $T_{1}$-complementary Hausdorff paratopological group topology, which gives a negative answer to \cite[Problem 10]{AT2017}. We give a very useful criterion for transversality in term of submaximal paratopological group topology, and prove that if a non-discrete paratopological group topology $G$ contains a central subgroup which admits a transversal paratopological group topology, then so does $G$. We introduce the concept of $PT$-sequence and give a characterization of an Abelian paratopological group being determined by a $PT$-sequence. As the applications, we prove that the Abelian paratopological group, which is endowed with the strongest paratopological group topology being determined by a $T$-sequence, does not admit a $T_{1}$-complementary Hausdorff paratopological group topology on $G$. Finally, we study the class of countable paratopological groups which is determined by a $PT$-filter, and obtain a sufficient condition for a countable paratopological group $G$ being determined by a $PT$-sequence which admits a transversal paratopological group topology on $G$ being determined by a $PT$-sequence.