Quiver with relations

A quiver with relations is a presentation of a $K$ -monoid as a quotient of the path algebra $K Γ ――$ of a (finite-vertex) quiver $Γ$ . Presenting an existing algebra $A$ this way is equivalent to finding a complete set of orthogonal idempotents $𝑒 1, \dots, 𝑒 𝑛$ such that $\sum 𝑛 𝑖 = 1 𝑒 𝑖 = 1$ and for each $𝑖, 𝑗 \in [𝑛]$ a set $Γ (𝑖, 𝑗) \subseteq A$ such that

{𝑒 𝑖} 𝑛 𝑖 = 1 \cup ⋃ 𝑖, 𝑗 \in [𝑛] Γ (𝑖, 𝑗)

generate $A$ .

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