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.



How to understand exchange pattern?
An explanation of important implications of these axioms happens directly after they are given in the paper. There importance is about the cluster dynamics (i.e. how things propagate from an initial seed). I will provide a few more details since the discussion in the paper is brief. 


The axiom E3 insures that the substitution $x\_j \leftarrow \frac{M\_0}{x\_j}$ in E4 replaces $x\_j$ with a monomial whenever the substitution actually happens. Note if at least one of $M\_j(t\_2)$ or $M\_j(t\_3)$ contain $x\_i$, then $M\_0$ is a monomial or zero since $x\_i = 0$ in $M\_0$. So, assume neither $M\_j(t\_2)$ or $M\_j(t\_3)$ involve $x\_i$, then by E3 applied to $t\_1 -^i t\_2 -^j t\_3$ and $t\_2 -^j t\_3 -^i t\_4$ neither $M\_i(t\_1)$ nor $M\_i(t\_2)$ involve $x\_j$ and the substitution never really happens.


Now Remark 2.2 deals with E4. The point is a we choose our initial data many things are determined. That is pick $t$ and then fix $M\_i(t)$ for $1 \leq i \leq n$ and fix $M\_j(t')$ for all $t -^j t'$. Then the ratios $M\_i(t')/M\_i(t'')$ whenever we have the situation $t -^j t' -^i t''$. To seen this ratio is determined look at $t''' -^i t -^j t' -^i t''$, then by E4
$$\frac{M\_i(t')}{M\_i(t'')} = \frac{M\_i(t)}{M\_i(t''')}|\_{x\_j \leftarrow M\_0/x\_j}$$
here the left side the ratio we claim is determined and the right side has the monomials we fixed. Also notice the monomials in $M\_0$ are among the monomials we fixed.


**Addition:** To address the question in the comments about the background of these axioms, the simple answer the to look at the absract of Fomin and Zelevinsky's *Cluster Algebra's I: Foundations.* The definitions and axoims for cluster algebras grew out of studying the dual canonical basis. From what I understand lots of examples (coordinate ring of $SL\_2$, Grassmannian, etc.) were looked at, the cluster algebra axoims seemed to be the "right" generalisation of the relations these examples satisfied. So, when you want to make the abstract technical definition of cluster algebras more concrete you can think of it as "generalised Plücker relations." Maybe this a not a satisfying answer. I know it wouldn't matter how long a stared at these examples, I would never have come out with the definition of a cluster algebra.


To see the cluster algebras aren't completely coming out of nowhere you can look at some of the papers in the references of *Cluster Algebra's I: Foundations.* For example in [Parametrizations of canonical bases and totally positive matrices](http://pages.uoregon.edu/arkadiy/bfz.pdf) by A. Berenstein, S. Fomin and A. Zelevinsky you will see pseudoline arrangements and other things reminiscent of cluster algebras.


