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Let $A$ be a preprojective algebra of Dynkin type.

Question 1: Let $P$ be an indecomposable projective $A$-module. It is known that the submodule lattice of $P$ is finite in Dynkin type $A_n$. Does this lattice have a concrete description or even a name? Has it been studied before?

Question 2: Is it known for which projective modules $P$, the submodule lattice of $P$ is finite?

Question 3: What is the best method (for example with QPA) to test whether the submodule lattice of a given module is distributive?

Let $A$ be a preprojective algebra of Dynkin type.

Question 1: Let $P$ be an indecomposable projective $A$-module. It is known that the submodule lattice of $P$ is finite. Does this lattice have a concrete description or even a name? Has it been studied before?

Question 2: Is it known for which projective modules $P$, the submodule lattice of $P$ is finite?

Question 3: What is the best method (for example with QPA) to test whether the submodule lattice of a given module is distributive?

Let $A$ be a preprojective algebra of Dynkin type.

Question 1: Let $P$ be an indecomposable projective $A$-module. It is known that the submodule lattice of $P$ is finite in Dynkin type $A_n$. Does this lattice have a concrete description or even a name? Has it been studied before?

Question 2: Is it known for which projective modules $P$, the submodule lattice of $P$ is finite?

Question 3: What is the best method (for example with QPA) to test whether the submodule lattice of a given module is distributive?

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Mare
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Submodule lattices of preprojective algebras

Let $A$ be a preprojective algebra of Dynkin type.

Question 1: Let $P$ be an indecomposable projective $A$-module. It is known that the submodule lattice of $P$ is finite. Does this lattice have a concrete description or even a name? Has it been studied before?

Question 2: Is it known for which projective modules $P$, the submodule lattice of $P$ is finite?

Question 3: What is the best method (for example with QPA) to test whether the submodule lattice of a given module is distributive?