  We introduce and study oscillator topologies on paratopological groups and
define certain related number invariants. As an application we prove that a
Hausdorff paratopological group $G$ admits a weaker Hausdorff group topology
provided $G$ is 3-oscillating. A paratopological group $G$ is 3-oscillating
(resp. 2-oscillating) provided for any neighborhood $U$ of the unity $e$ of $G$
there is a neighborhood $V\subset G$ of $e$ such that $V^{-1}VV^{-1}\subset
UU^{-1}U$ (resp. $V^{-1}V\subset UU^{-1}$). The class of 2-oscillating
paratopological groups includes all collapsing, all nilpotent paratopological
groups, all paratopological groups satisfying a positive law, all
paratopological SIN-group and all saturated paratopological groups (the latter
means that for any nonempty open set $U\subset G$ the set $U^{-1}$ has nonempty
interior). We prove that each totally bounded paratopological group $G$ is
countably cellular; moreover, every cardinal of uncountable cofinality is a
precaliber of $G$. Also we give an example of a saturated paratopological group
which is not isomorphic to its mirror paratopological group as well as an
example of a 2-oscillating paratopological group whose mirror paratopological
group is not 2-oscillating.
