Triangle equivalences between Gorenstein tiled orders and incidence algebras of posets

We prove that for any $\mathbb{N}$-graded Gorenstein tiled order $A$, the stable category ${\underset{‾}{\mathrm{C}\mathrm{M}}}^{\mathbb{Z}}A$ is triangle equivalent to the perfect derived category of the incidence algebra of a finite poset ${\mathbb{V}}_A^{op}$. Moreover, for a finite poset $P$, we prove that the incidence algebra of $P$ can be realized as the endomorphism algebra of a standard tilting object if and only if $P$ is either empty or has the maximum. We also study the behaviors of the corresponding poset under graded Morita equivalences and coverings of a Gorenstein tiled order. Finally, we classify Gorenstein tiled orders $A$ satisfying $\mathrm{\mid }{\mathbb{V}}_A^{op}\mathrm{\mid }\le 3$.