  In this paper, we apply the theory of inverse semigroups to the
$C^{*}$-algebra $U[\mathbb{Z}]$ considered in \cite{Cuntz}. We show that the
$C^{*}$-algebra $U[\mathbb{Z}]$ is generated by an inverse semigroup of partial
isometries. We explicity identify the groupoid $\mathcal{G}_{tight}$ associated
to the inverse semigroup and show that $\mathcal{G}_{tight}$ is exactly the
same groupoid obtained in \cite{Cuntz-Li}.
