New Hereditary and Mutation-Invariant Properties Arising from Forks

A hereditary property of quivers is a property preserved by restriction to any full subquiver. Similarly, a mutation-invariant property of quivers is a property preserved by mutation. Using forks, a class of quivers developed by M. Warkentin, we introduce a new hereditary and mutation-invariant property. We prove that a quiver being mutation-equivalent to a finite number of non-forks – defined as having a finite forkless part – is this new property, using only elementary methods. Additionally, we show that a more general property – having a finite pre-forkless part – is also a new hereditary and mutation-invariant property in much the same manner.