  Let $\Gamma^{+}$ be the positive cone of a totally ordered abelian discrete
group $\Gamma$, and $\alpha$ an action of $\Gamma^{+}$ by extendible
endomorphisms of a $C^*$-algebra $A$. We prove that the partial-isometric
crossed product $A\times_{\alpha}^{\textrm{piso}}\Gamma^{+}$ is a full corner
of a group crossed product $\mathcal{B}\times_{\beta}\Gamma$, where
$\mathcal{B}$ is a subalgebra of $\ell^{\infty}(\Gamma,A)$ generated by a
collection of faithful copies of $A$, and the action $\beta$ on $\mathcal{B}$
is induced by shift on $\ell^{\infty}(\Gamma,A)$. We then use this realization
to show that $A\times_{\alpha}^{\textrm{piso}}\Gamma^{+}$ has an essential
ideal $J$, which is a full corner in an ideal $\mathcal{I}\times_{\beta}\Gamma$
of $\mathcal{B}\times_{\beta}\Gamma$.
