Definition 2

Let C be a quiver. A path in C is a sequence $c= (a_0, f_1, a_1, . . . , a_{n-1}, f_n, a_n)$, where $n$ is an integer $\geqslant 0,a_0, . . . , a_n$ are vertices of C, and where, for $1\leqslant i\leqslant n,f_i$ is an arrow of C joining $a_{i-1}$ to $a_i$. One says that the path $c$ is of length $n$.

Let $c= (a_0, f_1, a_1, . . . , a_{n-1}, f_n, a_n)$ be a path of length $n$ in C. The vertex $a_0$ is called the origin, or the source, of $c$; the vertex $a_n$ is called the term, or the end, of $c$; one also says that $c$ joins the vertex $a_0$ to the vertex $a_n$. The vertex $a_i$, for $0\leqslant i\leqslant n$, is called the vertex of index $i$; the arrow $f_i$, for $1\leqslant i\leqslant n$, is called the $i$-th arrow, or the arrow of index $i$. A path of length $\geqslant 1$ is determined by the sequence of its arrows.

A path whose origin is equal to the term is called a loop. A path of length 0 is a loop, called constant.

One says that paths

$c= (a_0, f_1, a_1, . . . , a_{n-1}, f_n, a_n)$ and $c'= (a'_0, f'_1, a'_1, . . . , a'_{m-1}, f'_m, a'_m)$

in C are juxtaposed if the term $a_n$ of $c$ is the origin $a'_0$ of $c'$. In this case, the sequence $(a_0, f_1, a_1, . . . , a_{n-1}, f_n, a_n, f'_1, a'_1, . . . , f'_m, a'_m)$ is a path in C which is denoted by $c*c'$ and is called the juxtaposed path of $c$ and $c'$. It joins the origin of $c$ to the term of $c'$; its length is the sum of the lengths of the paths $c$ and $c'$.