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Let ${\cal O}_{*}$ be the C$^{*}$-algebra defined as the direct sum of allCuntz algebras. Then ${\cal O}_{*}$ has a non-cocommutative comultiplication$\Delta_{\phi}$ and a counit $\epsilon$. Let ${\rm BI}({\cal O}_{*})$ denotethe set of all closed biideals of the C$^{*}$-bialgebra $({\calO}_{*},\Delta_{\phi},\epsilon)$ and let ${\cal P}({\bf P})$ denote the powerset of the set of all prime numbers. We show a one-to-one correspondencebetween ${\rm BI}({\cal O}_{*})$ and ${\cal P}({\bf P})$. Furthermore, we showthat for any ${\cal I}$ in ${\rm BI}({\cal O}_{*})$, there exists aC$^{*}$-subbialgebra ${\cal B}_{{\cal I}}$ of ${\cal O}_{*}$ such that ${\calO}_{*}={\cal B}_{{\cal I}}\oplus {\cal I}$, and the set of all suchC$^{*}$-subbialgebras is a lattice with respect to the natural operations amongC$^{*}$-subbialgebras, which is isomorphic to the lattice ${\cal P}({\bf P})$.

Biideals and a lattice of C$^{*}$-bialgebras associated with prime numbers