Representation theory of the Boolean lattice

Question

Let $A_n$ be the incidence algebra of the Boolean lattice of an $n$-set.

Question 1: Is $A_3$ tame and if yes, is there a classification of all indecomposable modules for $A_3$?

Question 2: Let $\Omega^1(A_n)$ denote the full subcategory of modules that are submodules of a projective $A_n$-module. For which $n$ is $O_n=sup \{ t | \exists X :dim ( soc(X) ) =t , X \in \Omega^1(A_n)$ indecomposable $\}$ finite and what is the value of $O_n$ depending on $n$ ? (this is equivalent to the existence of a number $r$ such that every indecomposable module $X \in \Omega^1(A_n)$ embedds into $A^r$).

Answer 1

I only have a partial answer to Question 1. The algebra $A_3$ is tame. You can find a proof (due to Lenzing) in Lemma 3.5 of this article. The idea is: it is at least tame, since one can find a subposet of affine type. It is derived equivalent to the weighted projective line of type $(3,3,3)$ which is a tame hereditary category, hence it is at most tame.

I do not know if the indecomposable modules for $A_3$ have been classified by someone.