  A paratopological group $G$ has a {\it suitable set} $S$. The latter means
that $S$ is a discrete subspace of $G$, $S\cup \{e\}$ is closed, and the
subgroup $\langle S\rangle$ of $G$ generated by $S$ is dense in $G$. Suitable
sets in topological groups were studied by many authors. The aim of the present
paper is to provide a start-up for a general investigation of suitable sets for
paratopological groups, looking to what extent we can (by proving propositions)
or cannot (by constructing examples) generalize to paratopological groups
results which hold for topological groups, and to pose a few challenging
questions for possible future research. We shall discuss when paratopological
groups of different classes have suitable sets. Namely, we consider
paratopological groups (in particular, countable) satisfying different
separation axioms, paratopological groups which are compact-like spaces, and
saturated (in particular, precompact) paratopological groups. Also we consider
the permanence of a property of a group to have a suitable set with respect to
(open or dense) subgroups, products and extensions.
