Incidence algebras and beyond: brick-continuity and generic bricks

We will show that the twelve families of minimally representation-infinite incidence algebras are brick-continuous and admit a generic brick, which will be explicitly constructed. After that, reversing the reduction process between posets, we will show that any representation-infinite incidence algebra is brick-continuous and admits a generic brick, which can be explicitly computed using what will be called ``extension functors''. Lastly, a method to apply these results to a broader class of algebras is given.