Subgroups of products of paratopological groups

We give a characterization of the paratopological groups that can be topologically embedded as subgroups into a product of first-countable (second-countable) TiTi paratopological groups for i=0,1i=0,1. We show that a T1T1 paratopological group G   admits a topological embedding as a subgroup into a topological product of first-countable (second-countable) T1T1 paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the symmetry number of G is countable, i.e., for every neighborhood U of the identity e in G we can find a countable family γ of neighborhoods of e   satisfying ⋂V∈γV−1⊆U⋂V∈γV−1⊆U. We show that every 2-pseudocompact T1T1 paratopological group with a countable symmetry number is a topological group.We answer in the negative some questions posed by Manuel Sanchis and Mikhail Tkachenko by constructing an example of a commutative functionally Hausdorff totally ω-narrow paratopological group of countable pseudocharacter H such that there is no continuous isomorphism from H onto a Hausdorff first-countable paratopological group. The group H is not topologically isomorphic to a subgroup of a product of Hausdorff second-countable paratopological groups.