  Let $\Sigma$ be Laurent phenomenon (LP) seed of rank $n$,
$\mathcal{A}(\Sigma)$, $\mathcal{U}(\Sigma)$ and $\mathcal{L}(\Sigma)$ be its
corresponding Laurent phenomenon algebra, upper bound and lower bound
respectively. We prove that each seed of $\mathcal{A}(\Sigma)$ is uniquely
defined by its cluster, and any two seeds of $\mathcal{A}(\Sigma)$ with $n-1$
common cluster variables are connected with each other by one step of mutation.
The method in this paper also works for (totally sign-skew-symmetric) cluster
algebras. Moreover, we show that $\mathcal{U}(\Sigma)$ is invariant under seed
mutations when each exchange polynomials coincides with its exchange Laurent
polynomials of $\Sigma$. Besides, we obtain the standard monomial bases of
$\mathcal{L}(\Sigma)$. We also prove that $\mathcal{U}(\Sigma)$ coincides with
$\mathcal{L}(\Sigma)$ under certain conditions.
