Some elementary properties of Laurent phenomenon algebras

Let $\mathrm{\Sigma }$ be a Laurent phenomenon (LP) seed of rank $n$, $\mathcal{A}(\mathrm{\Sigma })$, $\mathcal{U}(\mathrm{\Sigma })$, and $\mathcal{L}(\mathrm{\Sigma })$ be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively. We prove that each seed of $\mathcal{A}(\mathrm{\Sigma })$ is uniquely defined by its cluster and any two seeds of $\mathcal{A}(\mathrm{\Sigma })$ 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 $\mathcal{U}(\mathrm{\Sigma })$ is invariant under seed mutations when each exchange polynomials coincides with its exchange Laurent polynomials of $\mathrm{\Sigma }$. Besides, we obtain the standard monomial bases of $\mathcal{L}(\mathrm{\Sigma })$. We also prove that $\mathcal{U}(\mathrm{\Sigma })$ coincides with $\mathcal{L}(\mathrm{\Sigma })$ under certain conditions.