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

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