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.
@article{ADM_2002_1_a2,
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},
year = {2002},
number = {1},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ADM_2002_1_a2/}
}
TY - JOUR AU - Zh. T. Chernousova AU - M. A. Dokuchaev AU - M. A. Khibina AU - V. V. Kirichenko AU - S. G. Miroshnichenko AU - V. N. Zhuravlev TI - Tiled orders over discrete valuation rings, nite Markov chains and partially ordered sets. I JO - Algebra and discrete mathematics PY - 2002 SP - 32 EP - 63 IS - 1 UR - http://geodesic.mathdoc.fr/item/ADM_2002_1_a2/ LA - en ID - ADM_2002_1_a2 ER -
%0 Journal Article %A Zh. T. Chernousova %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 %N 1 %U http://geodesic.mathdoc.fr/item/ADM_2002_1_a2/ %G en %F ADM_2002_1_a2
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/