The composition series of ideals of the partial-isometric crossed product by the semigroup $\mathbb{N}^{2}

Suppose that $\alpha$ is an action of the semigroup $\mathbb{N}^{2}$ on a $C^*$-algebra $A$ by endomorphisms. Let $A\times_{\alpha}^{\textrm{piso}} \mathbb{N}^{2}$ be the associated partial-isometric crossed product. By applying an earlier result which embeds this semigroup crossed product (as a full corner) in a crossed product by the group $\mathbb{Z}^{2}$, a composition series $0\leq L_{1}\leq L_{2}\leq A\times_{\alpha}^{\textrm{piso}} \mathbb{N}^{2}$ of essential ideals is obtained for which we identify the subquotients with familiar algebras.