  Let $\Gamma^{+}$ be the positive cone in a totally ordered abelian group
$\Gamma$, and $\alpha$ an action of $\Gamma^{+}$ by extendible endomorphisms of
a $C^{\ast}$-algebra $A$. Suppose $I$ is an extendible $\alpha$-invariant ideal
of $A$. We prove that the partial-isometric crossed product
$\mathcal{I}:=I\times_{\alpha}^{\textrm{piso}}\Gamma^{+}$ embeds naturally as
an ideal of $A\times_{\alpha}^{\textrm{piso}}\Gamma^{+}$, such that the
quotient is the partial-isometric crossed product of the quotient algebra. We
claim that this ideal $\mathcal{I}$ together with the kernel of a natural
homomorphism $\phi: A\times_{\alpha}^{\textrm{piso}}\Gamma^{+}\rightarrow
A\times_{\alpha}^{\textrm{iso}}\Gamma^{+}$ gives a composition series of ideals
of $A\times_{\alpha}^{\textrm{piso}}\Gamma^{+}$ studied by Lindiarni and
Raeburn.
