Cellularity in subgroups of paratopological groups

It is known that the cellularity of every σ-compact paratopological group is countable, without assuming any separation restrictions on the group. We prove that every subgroup of a σ  -compact T1T1 paratopological group has countable cellularity, but this conclusion fails for subgroups of σ  -compact T0T0 paratopological groups. For every infinite cardinal κ, we present a σ-compact subsemigroup H of a Hausdorff topological group such that the cellularity of H equals κ.We also prove that if S is a non-empty subsemigroup of a topologically periodic semitopological group G, then the closure of S is a subgroup of G. This implies, in particular, that the closure of every non-empty subsemigroup of a precompact topological group G is a subgroup of G and that every subsemigroup of G has countable cellularity.