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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 $T_{1}$-complementaryHausdorff paratopological group topology, which gives a negative answer to\cite[Problem 10]{AT2017}. We give a very useful criterion for transversalityin term of submaximal paratopological group topology, and prove that if anon-discrete paratopological group topology $G$ contains a central subgroupwhich admits a transversal paratopological group topology, then so does $G$. Weintroduce the concept of $PT$-sequence and give a characterization of anAbelian paratopological group being determined by a $PT$-sequence. As theapplications, we prove that the Abelian paratopological group, which is endowedwith the strongest paratopological group topology being determined by a$T$-sequence, does not admit a $T_{1}$-complementary Hausdorff paratopologicalgroup topology on $G$. Finally, we study the class of countable paratopologicalgroups which is determined by a $PT$-filter, and obtain a sufficient conditionfor a countable paratopological group $G$ being determined by a $PT$-sequencewhich admits a transversal paratopological group topology on $G$ beingdetermined by a $PT$-sequence.

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