Weak completions of paratopological groups

Given a $T_0$ paratopological group $G$ and a class $mathcal C$ of continuous homomorphisms of paratopological groups, we define the $mathcal C$-$semicompletion$ $mathcal C[G)$ and $mathcal C$-$completion$ $mathcal C[G]$ of the group $G$ that contain $G$ as a dense subgroup, satisfy the $T_0$-separation axiom and have certain universality properties. For special classes $mathcal C$, we present some necessary and sufficient conditions on $G$ in order that the (semi)completions $mathcal C[G)$ and $mathcal C[G]$ be Hausdorff. Also, we give an example of a Hausdorff paratopological abelian group $G$ whose $mathcal C$-semicompletion $mathcal C[G)$ fails to be a $T_1$-space, where $mathcal C$ is the class of continuous homomorphisms of sequentially compact topological groups to paratopological groups. In particular, the group $G$ contains an $omega$-bounded sequentially compact subgroup $H$ such that $H$ is a topological group but its closure in $G$ fails to be a subgroup.