Woah! And for the exact same reasons too... Thanks for pointing me towards that paper! It doesn't answer the question, but it shows that I'm in good company looking at these things. :)

Yep... that's just what Wikipedia (and others?) call these identities. Maybe if someone understands the geometry underlying these identities, they can suggest a better name!

Yes, that's what I mean. It happens sometimes. E.g., take a zigzag quiver with even numbers pointing to odd, like 0 -> 1 <- 2 -> 3 <- 4 -> 5 <- 6 -> 7 <- 8. Take an indec module like the interval module [2,6]. I think the irreducible morphisms starting at [2,6] are all epis... they have targets [2,4] and [4,6]. At the same time [2,6] injects into [2,4] \oplus [4,6].

@StephenMcKean Oh right -- this is getting kind of neat! Let me know if you end up looking at subsets of the torus computationally. It could be an interesting student project too. Feel free to drop me an email if you compute things or if you want me to ask some students.