  The dynamics of a one-sided subshift $\mathsf{X}$ can be modeled by a set of
partially defined bijections. From this data we define an inverse semigroup
$\mathcal{S}_{\mathsf{X}}$ and show that it has many interesting properties. We
prove that the Carlsen-Matsumoto C*-algebra $\mathcal{O}_\mathsf{X}$ associated
to $\mathsf{X}$ is canonically isomorphic to Exel's tight C*-algebra of
$\mathcal{S}_{\mathsf{X}}$. As one consequence, we obtain that
$\mathcal{O}_\mathsf{X}$ can be written as a partial crossed product of a
commutative C*-algebra by a countable group.
