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

Let ${\mathrm{\Gamma }}^{+}$ be the positive cone in a totally ordered abelian group $\mathrm{\Gamma }$, and $\alpha$ an action of ${\mathrm{\Gamma }}^{+}$ 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 $\mathcal{I}:=I{\times }_{\alpha }^{\text{piso}}{\mathrm{\Gamma }}^{+}$ embeds naturally as an ideal of $A{\times }_{\alpha }^{\text{piso}}{\mathrm{\Gamma }}^{+}$, 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 $\phi: A\times_{\alpha}^{$\text{piso}$}\Gamma^{+}\rightarrow A\times_{\alpha}^{$\text{iso}$}\Gamma^{+}$ gives a composition series of ideals of $A{\times }_{\alpha }^{\text{piso}}{\mathrm{\Gamma }}^{+}$ studied by Lindiarni and Raeburn.