  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.
