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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