Geodesic
Tiled orders over discrete valuation rings, nite Markov chains and partially ordered sets.~I
Algebra and discrete mathematics, no. 1 (2002), pp. 32-63.

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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. 
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     author = {Zh. T. Chernousova and M. A. Dokuchaev and M. A. Khibina and V. V. Kirichenko and S. G. Miroshnichenko and V. N. Zhuravlev},
     title = {Tiled orders over discrete valuation rings, nite {Markov} chains and partially ordered {sets.~I}},
     journal = {Algebra and discrete mathematics},
     pages = {32--63},
     publisher = {mathdoc},
     number = {1},
     year = {2002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2002_1_a2/}
}
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%A M. A. Dokuchaev
%A M. A. Khibina
%A V. V. Kirichenko
%A S. G. Miroshnichenko
%A V. N. Zhuravlev
%T Tiled orders over discrete valuation rings, nite Markov chains and partially ordered sets.~I
%J Algebra and discrete mathematics
%D 2002
%P 32-63
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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 and discrete mathematics, no. 1 (2002), pp. 32-63. http://geodesic.mathdoc.fr/item/ADM_2002_1_a2/