  We propose two universal constructions of globalization of a partial action
of a semigroup on a set, satisfying certain conditions which arise in Morita
theory of semigroups. One of the constructions is based on the tensor product
of a partial semigroup act with the semigroup and generalizes the globalization
construction of strong partial actions of monoids due to Megrelishvili and
Schr\"oder. It produces the initial object in an appropriate caterory of
globalizations of a given partial action. The other construction involves
${\mathrm{Hom}}$-sets and is novel even in the monoid setting. It produces the
terminal object in an appropriate category of globalizations. While in the
group case the results of the two constructions are isomorphic, they can be far
different in the monoid case.
