P$ over $P$ are [*equivalent*]{} if there is a weak $T^n$- equivariant homeomorphism $\phi :M\to N$ (i.e., there exists an automorphism $\rho$ on $T^n$ such that $\phi(tx)=\rho(t)\phi(x)$) such that $\pi_N\circ\phi=\pi_M$.

Let $P$ be an $n$-dimensional simple polytope with $m$ facets, and let $M$ be a quasitoric manifold over $P$. Then we can find a characteristic map $\chi$ for $M$ such that $\chi(F_1)=(1,0,\ldots,0),\ldots, \chi(F_n)=(0,\ldots,0,1)$ where $F_1,\ldots, F_n$ are the facets meeting at one particular vertex $p\in P$. Then we can define an $(m-n)\times n$ matrix $A$ whose row vectors are $\chi(F_{n+1}), \ldots, \chi(F_{m})$. This matrix $A$ is called a [*characteristic matrix*]{} of $M$. For the details about quasitoric manifolds we refer the reader to [@DJ]. We note that a Bott manifold $B_n$ associated with the matrix $\Lambda$ in admits the canonical nice $T^n$-action with which $B_n$ becomes a quasitoric manifold. The characteristic matrix of $B_n$ is then equal to $-\Lambda - I_n$, where $I_n$ is the identity matrix of size $n$, see [@MP] for details.

In this section we will consider quasitoric manifolds whose cohomology rings resemble those of Bott manifolds. For this we need the following definition.

\[def:BQ-algebra\] A graded algebra $S$ over $\mathbb Z$ generated by $x_1,\ldots,x_n$ of degree 