  For operators $A$, it is sometimes possible to define $e^{At}$ as an operator
in and of itself provided it meets certain regularity conditions. Like
$e^{\lambda x}$ for ODEs, this operator is useful for solving PDEs involving
the operator $A$. We call the set of $e^{At}$ a semigroup generated by $A$. In
this paper, we discuss the properties of semigroups generated by the fractional
integral, an operator appearing in PDEs in increasingly many fields, over
Bochner-Lebesgue spaces.
