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

The main concept of this part of the paper is that of a reduced exponent matrix and its quiver, which is strongly connected and simply laced. We give the description of quivers of reduced Gorenstein exponent matrices whose number \(s\) of vertices is at most \(7\). For \(2\leq s\leq 5\) we have that...

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Date:	2018
Main Authors:	Chernousova, Zh. T., Dokuchaev, M. A., Khibina, M. A., Kirichenko, V. V., Miroshnichenko, S. G., Zhuravlev, V. N.
Format:	Article
Language:	English
Published:	Lugansk National Taras Shevchenko University 2018
Subjects:	semiperfect ring exponent matrix tiled order quiver partially ordered set index of semiperfect ring Gorenstein tiled order global dimension transition matrix 16P40 16G10
Online Access:	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/958
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Journal Title:	Algebra and Discrete Mathematics

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Algebra and Discrete Mathematics

id	oai:ojs.admjournal.luguniv.edu.ua:article-958
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spelling	oai:ojs.admjournal.luguniv.edu.ua:article-9582018-05-13T07:14:40Z Tiled orders over discrete valuation rings, finite Markov chains and partially ordered sets. II Chernousova, Zh. T. Dokuchaev, M. A. Khibina, M. A. Kirichenko, V. V. Miroshnichenko, S. G. Zhuravlev, V. N. semiperfect ring, exponent matrix, tiled order, quiver, partially ordered set, index of semiperfect ring, Gorenstein tiled order, global dimension, transition matrix 16P40, 16G10 The main concept of this part of the paper is that of a reduced exponent matrix and its quiver, which is strongly connected and simply laced. We give the description of quivers of reduced Gorenstein exponent matrices whose number \(s\) of vertices is at most \(7\). For \(2\leq s\leq 5\) we have that all adjacency matrices of such quivers are multiples of doubly stochastic matrices. We prove that for any permutation \(\sigma \) on \(n\) letters without fixed elements there exists a reduced Gorenstein tiled order \(\Lambda\) with \(\sigma ({\mathcal{E}})=\sigma\). We show that for any positive integer \(k\) there exists a Gorenstein tiled order \(\Lambda_{k}\) with \(in\Lambda_{k}=k\). The adjacency matrix of any cyclic Gorenstein order \(\Lambda \) is a linear combination of powers of a permutation matrix \(P_{\sigma}\) with non-negative coefficients, where \(\sigma = \sigma(\Lambda)\). If \(A\) is a noetherian prime semiperfect semidistributive ring of a finite global dimension, then \(Q(A)\) be a strongly connected simply laced quiver which has no loops. Lugansk National Taras Shevchenko University 2018-05-13 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/958 Algebra and Discrete Mathematics; Vol 2, No 2 (2003) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/958/487 Copyright (c) 2018 Algebra and Discrete Mathematics
institution	Algebra and Discrete Mathematics
baseUrl_str
datestamp_date	2018-05-13T07:14:40Z
collection	OJS
language	English
topic	semiperfect ring exponent matrix tiled order quiver partially ordered set index of semiperfect ring Gorenstein tiled order global dimension transition matrix 16P40 16G10
spellingShingle	semiperfect ring exponent matrix tiled order quiver partially ordered set index of semiperfect ring Gorenstein tiled order global dimension transition matrix 16P40 16G10 Chernousova, Zh. T. Dokuchaev, M. A. Khibina, M. A. Kirichenko, V. V. Miroshnichenko, S. G. Zhuravlev, V. N. Tiled orders over discrete valuation rings, finite Markov chains and partially ordered sets. II
topic_facet	semiperfect ring exponent matrix tiled order quiver partially ordered set index of semiperfect ring Gorenstein tiled order global dimension transition matrix 16P40 16G10
format	Article
author	Chernousova, Zh. T. Dokuchaev, M. A. Khibina, M. A. Kirichenko, V. V. Miroshnichenko, S. G. Zhuravlev, V. N.
author_facet	Chernousova, Zh. T. Dokuchaev, M. A. Khibina, M. A. Kirichenko, V. V. Miroshnichenko, S. G. Zhuravlev, V. N.
author_sort	Chernousova, Zh. T.
title	Tiled orders over discrete valuation rings, finite Markov chains and partially ordered sets. II
title_short	Tiled orders over discrete valuation rings, finite Markov chains and partially ordered sets. II
title_full	Tiled orders over discrete valuation rings, finite Markov chains and partially ordered sets. II
title_fullStr	Tiled orders over discrete valuation rings, finite Markov chains and partially ordered sets. II
title_full_unstemmed	Tiled orders over discrete valuation rings, finite Markov chains and partially ordered sets. II
title_sort	tiled orders over discrete valuation rings, finite markov chains and partially ordered sets. ii
description	The main concept of this part of the paper is that of a reduced exponent matrix and its quiver, which is strongly connected and simply laced. We give the description of quivers of reduced Gorenstein exponent matrices whose number \(s\) of vertices is at most \(7\). For \(2\leq s\leq 5\) we have that all adjacency matrices of such quivers are multiples of doubly stochastic matrices. We prove that for any permutation \(\sigma \) on \(n\) letters without fixed elements there exists a reduced Gorenstein tiled order \(\Lambda\) with \(\sigma ({\mathcal{E}})=\sigma\). We show that for any positive integer \(k\) there exists a Gorenstein tiled order \(\Lambda_{k}\) with \(in\Lambda_{k}=k\). The adjacency matrix of any cyclic Gorenstein order \(\Lambda \) is a linear combination of powers of a permutation matrix \(P_{\sigma}\) with non-negative coefficients, where \(\sigma = \sigma(\Lambda)\). If \(A\) is a noetherian prime semiperfect semidistributive ring of a finite global dimension, then \(Q(A)\) be a strongly connected simply laced quiver which has no loops.
publisher	Lugansk National Taras Shevchenko University
publishDate	2018
url	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/958
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first_indexed	2025-07-17T10:36:49Z
last_indexed	2025-07-17T10:36:49Z
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Tiled orders over discrete valuation rings, finite Markov chains and partially ordered sets. II

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