Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: astro-ph
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The categories of this paper are: physics
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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: cond-mat
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The categories of this paper are: atom-ph
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The categories of this paper are: eess
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The categories of this paper are: q-alg
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The categories of this paper are: stat
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The categories of this paper are: comp-gas
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The categories of this paper are: plasm-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: chem-ph