  The circular peak set of a permutation $\sigma$ is the set $\{\sigma(i)\mid
\sigma(i-1)<\sigma(i)>\sigma(i+1)\}$. Let $\mathcal{P}_n$ be the set of all the
subset $S\subseteq [n]$ such that there exists a permutation $\sigma$ which has
the circular set $S$. We can make the set $\mathcal{P}_n$ into a poset
$\mathscr{P}_n$ by defining $S\preceq T$ if $S\subseteq T$ as sets. In this
paper, we prove that the poset $\mathscr{P}_n$ is a simplicial complex on the
vertex set $[3,n]$. We study the $f$-vector, the $f$-polynomial, the reduced
Euler characteristic, the M$\ddot{o}$bius function, the $h$-vector and the
$h$-polynomial of $\mathscr{P}_n$. We also derive the zeta polynomial of
$\mathscr{P}_n$ and give the formula for the number of the chains in
$\mathscr{P}_n$. By the poset $\mathscr{P}_n$, we define two algebras
$\mathcal{A}_{\mathscr{P}_n}$ and $\mathcal{B}_{\mathscr{P}_n}$. We consider
the Hilbert polynomials and the Hilbert series of the algebra
$\mathcal{A}_{\mathscr{P}_n}$ and $\mathcal{B}_{\mathscr{P}_n}$.
