I am reading an paper "cluster algebras I: foundations" by Fomin and Zelevinsky.
Let $I = \{1,2, \ldots, n\}$ and $\mathbf{x}$ a cluster.
For each $t \in \mathbb{T}_n$, let $\mathbf{x}(t) = (x_i(t))_{i \in I}$. All variables will commute and satisfy the following exchange relations, for $t \overset{j}{-} t'$ in $\mathbb{T}_n$: \begin{align} & x_i(t) = x_i (t') \textrm{ for any $i \neq j$}; \\ & x_j (t)x_j (t') = M_j(t)(\mathbf{x}(t)) + M_j(t')(\mathbf{x}(t')). \end{align}
$\mathbf{Definition 2.1}$ An exchange pattern on $\mathbb{T}_n$ with coefficients in $\mathbb{P}$ is a family of monomials $\mathcal{M} = (M_j(t))_{t \in \mathbb{T}_n, j \in I}$ of the form \begin{align*} M_j(t) = p_j(t) \prod_{i \in I} x_i^{b_i}, \ p_j(t) \in \mathbb{P}, \ b_i \in \mathbb{Z}_{\geq 0}, \end{align*} satistfying the following four axioms:
E1. If $t \in \mathbb{T}_n$, then $x_j \not \mid M_j (t)$.
E2. If $t_1 \overset{j}{-} t_2$ and $x_i \mid M_j (t_1)$, then $x_i \not \mid M_j (t_2)$.
E3. If $t_1 \overset{i}{-} t_2 \overset{j}{-} t_3$, then $x_j \mid M_i (t_1)$ if and only if $x_i \mid M_j (t_2)$.
E4. Let If $t_1 \overset{i}{-} t_2 \overset{j}{-} t_3 \overset{i}{-} t_4$. Then $\frac{M_i(t_3)}{M_i(t_4)} = \frac{M_i(t_2)}{M_i(t_1)}|_{x_j \leftarrow \frac{M_{0}}{x_j}}$, where $M_0 = (M_j(t_2) + M_j(t_3))|_{x_i = 0}$.
E2 $\Rightarrow$ $M_j (t_1)$ and $M_j (t_2)$ are two monomials without common divisors in $ \mathbf{x} - {x_j}$. I do not know what E3 and E4 mean.