Window Posets of Graded Gorenstein Tiled Orders and a Question of Iyama and Liu

We study the window posets associated with basic N-graded Gorenstein tiled orders and address Question 5.7 posed by Iyama and Liu concerning hereditary incidence algebras. We establish a combinatorial description of these posets in terms of exponent matrices and b-invariants. We prove that if the associated incidence algebra is hereditary and at least one b-invariant is greater than 2, then the poset must be of type A whenever all b-invariants are at least 2 or some b-invariant is at least 4. We construct counterexamples for every size n ≥ 3 with b-invariants (1, 3, 2, ..., 2), whose Hasse diagrams are stars, demonstrating that the general answer to Iyama and Liu's question is negative. We also obtain a necessary and sufficient criterion for heredity when all b-invariants are at least 2 and derive a counting formula for the number of elements of the associated poset. These results clarify the relationship between Gorenstein tiled orders, hereditary incidence algebras, and representation types.