Let $Q = (Q_0, Q_1)$ be a quiver, and pick some $i \in Q_0$. Define the quiver representation $M$ by $$M_j = \begin{cases} k & \text{ if there is a path from $i$ to $j$,} \\ 0 & \text{ otherwise;} \end{cases}$$ for each $j \in Q_0$, and $$\varphi_\alpha = \begin{cases} 1_k & \text{ if } M_{s(\alpha)} = M_{t(\alpha)} = k,\\ 0 & \text{ otherwise.} \end{cases}$$
So for instance if $$Q = \left(\begin{array}{ccccc} 1 & \rightarrow & 2 & \rightarrow & 3 \\ & & \downarrow & & \downarrow\\ & & 4 & \rightarrow & 5 \end{array}\right)$$
then $M_4$ would be the representation
$$\begin{array}{ccccc} 0 & \rightarrow & 0 & \rightarrow & 0 \\ & & \downarrow & & \downarrow\\ & & k & = & k \end{array}$$
Does this representation have a name?
(Actually, I'm only interested in the case where $Q$ has no directed cycle, so if there's something that specializes to this in that case that has a name I'd be interested in that as well.)