The composition series of ideals of the partial-isometric crossed product by semigroup of endomorphisms

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.