Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The composition series of ideals of the partial-isometric crossed
  product by the semigroup $\mathbb{N}^{2}$   Suppose that $\mathbb{N}^{2}$ is the positive cone of the abelian
lattice-ordered group $\mathbb{Z}^{2}$, and $\alpha$ an action of
$\mathbb{N}^{2}$ by extendible endomorphisms of a $C^*$-algebra $A$. Let
$A\times_{\alpha}^{\textrm{piso}} \mathbb{N}^{2}$ be the partial-isometric
crossed product by the endomorphic action $\alpha$. 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.

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The categories of this paper are: astro-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-bio
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The categories of this paper are: acc-phys
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The categories of this paper are: dg-ga
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The categories of this paper are: mtrl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: gr-qc
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The categories of this paper are: eess
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The categories of this paper are: hep-th
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The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: funct-an
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The categories of this paper are: cs