$Q$ with target (source) $k$. Next $\mu_k(Q)$ is obtained from $Q$ by

-   adding a new arrow $i\to j$ for every existing pair of arrows $i\to k$ and $k\to j$;

-   reversing the orientation of every arrow with target or source equal to $k$;

-   erasing every pair of opposite arrows possibly created by (a).

It is easy to check that $\mu_k(\Si)$ is a seed, and $\mu_k(\mu_k(\Si)) = \Si$. The *mutation class* $\CC(\Si)$ is the set of all seeds obtained from $\Si$ by a finite sequence of mutations $\mu_k$. One can think of the elements of $\CC(\Si)$ as the vertices of an $n$-regular tree in which every edge stands for a mutation. If $\Si' = ((y'_1,\ldots,y'_n),Q')$ is a seed in $\CC(\Si)$, then the subset $\{y'_1,\ldots,y'_n\}$ is called a *cluster*, and its elements are called *cluster variables*.

Now, Fomin and Zelevinsky define the *cluster algebra* $\A_\Si$ as the subring of $\F$ generated by all cluster variables. Some important elements of $\A_\Si$ are the *cluster monomials*, monomials in the cluster variables supported on a single cluster.

\[ex2.1\] If $n=2$ and $\Si=((x_1,x_2),Q)$, where $Q$ is the quiver with $a$ arrows from $1$ to $2$, then $\A_\Si$ is the subring of $\Q(x_1,x_2)$ generated by 