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Some elementary properties of Laurent phenomenon algebras

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