Definition 1

Let C be a quiver. Let J denote the set of pairs $(f, g)$ of arrows of C such that $t(f) =o(g)$. A law of composition in C is a mapping $m: J\rightarrow$ Fl(C) such that, for every pair $(f, g)\in J$, the origin of the arrow $m(f, g)$ is that of $f$ and its end that of $g$.

Let C be a quiver endowed with a law of composition $m$. Two arrows $f$ and $g$ such that $t(f) =o(g)$ are said to be composable; the arrow $m(f, g)$ is called their composite, or their product, and is often denoted by $f\cdot g$, or even by $f g$.

For every vertex $a$ of C, the set $C_{a,a}$ = Fl$_{a,a}(C)$, endowed with the mapping deduced from $m$ by passing to subsets, is a magma (A, I, p. 1, Def. 1).

One says that a family $(f_i)_{i\in I}$ of arrows of C, indexed by a finite nonempty totally ordered set I, is composable if, for every pair $(i, j)$ of consecutive elements of I, the arrows $f_i$ and $f_j$ are composable. The product $\prod_{i\in I}f_i$ of such a family is defined by induction on the cardinal of I in the following way (cf. A, I, p. 3):

(i) if I has a single element $\omega ,\prod_{i\in I}f_i=f_{\omega}$;

(ii) if I has at least two elements and if $\omega$ is the smallest element of I, one sets $\prod_{i\in I}f_i=f_{\omega}\cdot \prod_{i\in I-\{\omega\}}f_i$.

The law of composition $m$ is said to be associative if one has $f\cdot (g\cdot h) =$ $(f\cdot g)\cdot h$ for every composable triple $(f, g, h)$ of arrows of C. When the law $m$ is associative, the product of a sequence $(f_1, . . . , f_n)$ of $n$ composable arrows, where $n$ is an integer $\geqslant 1$, is denoted by $f_1f_2. . . f_n$.

Let $a$ be a vertex of C. One says that an arrow $e\in C_{a,a}$ is an identity element of $m$ at $a$ if one has $ef=f$ for every arrow $f$ with origin $a$ and $ge=g$ for every arrow $g$ with end $a$. There exists at most one identity element of $m$ at $a:$ if $e$ and $e'$ are two such elements, one has $e'=ee'=e$. When such an identity element exists, it is often denoted by $e_a$; it is then an identity element of the magma $C_{a,a}$ (A, I, p. 12).