Paths

Recall

Formal definition cited in this section, gathered here for quick reference (full citation in the Bibliography):

Path. “A path of length $l \geq 1$ with source $a$ and target $b$ … is a sequence $(a \mid \alpha_1, \alpha_2, \ldots, \alpha_l \mid b)$ … We also agree to associate with each point $a \in Q_0$ a path of length $l = 0$, called the trivial or stationary path at $a$, and denoted by $\varepsilon_a = (a | a)$” (AssemSimsonSkowronski2006, Ch. II §1, p. 42–43, unnumbered definition immediately preceding Definition 1.2 — which itself defines the path algebra built from these paths, not the path concept).

A path in $Q$ is a sequence of arrows that can be composed head-to-tail: $\alpha_1, \alpha_2, \dots, \alpha_n$ with $t(\alpha_i) = s(\alpha_{i+1})$ for each consecutive pair. In addition, for each vertex $i$, a trivial path $e_i$ of length $0$ is allowed, starting and ending at $i$ and composing with nothing but itself.

In the running example, the paths are: $e_1, e_2, e_3$ (trivial), $\alpha$, $\beta$, and $\beta\alpha$ (first $\alpha$, then $\beta$), and nothing else, since there is no arrow out of $3$ or into $1$.

References

Full citations in the Bibliography. Formal definitions are gathered in Recall, above.

Assem, Simson, and Skowroński (AssemSimsonSkowronski2006), Ch. II §1, pp. 42–43, unnumbered definition preceding Definition 1.2 — the path definition itself is this unnumbered text; Definition 1.2, a few lines later, defines the path algebra $KQ$ built from these paths, not the path concept (an earlier draft of this book mislabeled the path definition itself as “Definition 1.2”).
Schiffler (Schiffler2014), Definition 2.1 and Example 2.2 (Chapter 2, §2.1) — same notion, called the “constant path” (or “lazy path”) $e_i$ at vertex $i$ for the length-$0$ case.

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