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Algebra and Discrete Mathematics, 2002, Issue 1, Pages 32–63
(Mi adm398)
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This article is cited in 7 scientific papers (total in 7 papers)
RESEARCH ARTICLE
Tiled orders over discrete valuation rings, nite Markov chains and partially ordered sets. I
Zh. T. Chernousovaa, M. A. Dokuchaevb, M. A. Khibinac, V. V. Kirichenkoa, S. G. Miroshnichenkoa, V. N. Zhuravleva a Faculty of Mechanics and Mathematics,
Kiev National Taras Shevchenko Univ.,
Vladimirskaya Str., 64, Kiev, Ukraine
b Departamento de Matematica Univ. de Sao
Paulo, Caixa Postal 66281, Sao Paulo, SP,
05315–970 — Brazil
c Glushkov In-t of Cybernetics NAS Ukraine,
Glushkov Av., 40, 03680 Kiev, Ukraine
Abstract:
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 $\Lambda$ is integral if $in\,\Lambda$ is an integer. Every cyclic Gorenstein tiled order is integral. In particular, $in\, \Lambda\,=\,1$ if and only if $\Lambda$ 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 $in\,\Lambda\,=\,w(\Lambda )\,=\,1$, or $in\,\Lambda\,=\,w(\Lambda )\,=\,2$, where $w(\Lambda )$ is a width of $\Lambda$.
Keywords:
semiperfect ring, tiled order, quiver, partially ordered set, index of semiperfect ring, Gorenstein tiled order, finite Markov chain.
Received: 26.10.2002
Citation:
Zh. T. Chernousova, M. A. Dokuchaev, M. A. Khibina, V. V. Kirichenko, S. G. Miroshnichenko, V. N. Zhuravlev, “Tiled orders over discrete valuation rings, nite Markov chains and partially ordered sets. I”, Algebra Discrete Math., 2002, no. 1, 32–63
Linking options:
https://www.mathnet.ru/eng/adm398 https://www.mathnet.ru/eng/adm/y2002/i1/p32
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