Piecewise Hereditary algebras of Dynkin and extended Dynkin type

We present a study on the description of incidence algebras that are piecewise hereditary, which we denominate Phia algebras. We describe the quiver with relations of the Phia algebras of Dynkin type and introduce a new family of Phia algebras of extended Dynkin type, which we call ANS family, in reference to Assem, Nehring, and Skowro\'nski. In this description, the important method was the one of cutting sets on trivial extensions, inspired by this we made of a computer program which shows exactly the cutting sets on the given trivial extension that result on incidence algebras.