  Certain $*$-semigroups are associated with the universal $C^*$-algebra
generated by a partial isometry, which is itself the universal $C^*$-algebra of
a $*$-semigroup. A fundamental role for a $*$-structure on a semigroup is
emphasized, and ordered and matricially ordered $*$-semigroups are introduced,
along with their universal $C^*$-algebras. The universal $C^*$-algebra
generated by a partial isometry is isomorphic to a relative Cuntz-Pimsner
$C^*$-algebra of a $C^*$-correspondence over the $C^*$-algebra of a matricially
ordered $*$-semigroup. One may view the $C^*$-algebra of a partial isometry as
the crossed product algebra associated with a dynamical system defined by a
complete order map modelled by a partial isometry acting on a matricially
ordered $*$-semigroup.
