Tiled orders over discrete valuation rings, finite Markov chains and partially ordered sets. I

• Опубліковано в: : Algebra and Discrete Mathematics
• Дата: 2002
• Автори: Chernousova, Zh.T., Dokuchaev, M.A., Khibina, M.A., Kirichenko, V.V., Miroshnichenko, S.G., Zhuravlev, V.N.
• Формат: Стаття
• Мова: Англійська
• Опубліковано: Інститут прикладної математики і механіки НАН України 2002
• Онлайн доступ: https://nasplib.isofts.kiev.ua/handle/123456789/155280
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: Tiled orders over discrete valuation rings, finite Markov chains and partially ordered sets. I / Zh.T. Chernousova, M.A. Dokuchaev, M.A. Khibina, V.V. Kirichenko, S.G. Miroshnichenko, V.N. Zhuravlev // Algebra and Discrete Mathematics. — 2002. — Vol. 1, № 1. — С. 32–63. — назв. — англ.

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.