Tiled orders over discrete valuation rings, finite Markov chains and partially ordered sets: I
Zhana T. Chernousova, Mikhailo Dokuchaev, Marina A. Khibina, Vladimir Vasil'evich Kirichenko, Svetlane G. Miroshnichenko, V. N. Zhuravlev · The scientific electronic library of periodicals of the National Academy of Sciences of Ukraine (National Academy of Sciences of Ukraine) · 2002
We prove that the quiver of tiled order over a discrete valuation ring is strongly connected and simply laced. With such quiver we associate a finite ergodic Markov chain. We introduce the notion of the index in A of a right noetherian semiperfect ring A as the maximal real eigen-value of its adjacency matrix. A tiled order Λ is integral if in Λ is an integer. Every cyclic Gorenstein tiled order is integral. In particular, in Λ = 1 if and only if Λ is hereditary. We give an example of a non-integral Gorenstein tiled order. We prove that a reduced (0, 1)-order is Gorenstein if and only if either inΛ = w(Λ) = 1, or inΛ = w(Λ) = 2, where w(Λ) is a width of Λ.