Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Biideals and a lattice of C$^{*}$-bialgebras associated with prime
  numbers   Let ${\cal O}_{*}$ be the C$^{*}$-algebra defined as the direct sum of all
Cuntz algebras. Then ${\cal O}_{*}$ has a non-cocommutative comultiplication
$\Delta_{\phi}$ and a counit $\epsilon$. Let ${\rm BI}({\cal O}_{*})$ denote
the set of all closed biideals of the C$^{*}$-bialgebra $({\cal
O}_{*},\Delta_{\phi},\epsilon)$ and let ${\cal P}({\bf P})$ denote the power
set of the set of all prime numbers. We show a one-to-one correspondence
between ${\rm BI}({\cal O}_{*})$ and ${\cal P}({\bf P})$. Furthermore, we show
that for any ${\cal I}$ in ${\rm BI}({\cal O}_{*})$, there exists a
C$^{*}$-subbialgebra ${\cal B}_{{\cal I}}$ of ${\cal O}_{*}$ such that ${\cal
O}_{*}={\cal B}_{{\cal I}}\oplus {\cal I}$, and the set of all such
C$^{*}$-subbialgebras is a lattice with respect to the natural operations among
C$^{*}$-subbialgebras, which is isomorphic to the lattice ${\cal P}({\bf P})$.

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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: q-alg
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: adap-org
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The categories of this paper are: funct-an
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The categories of this paper are: quant-ph
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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: alg-geom
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The categories of this paper are: econ
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The categories of this paper are: bayes-an