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

• 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
• Journal Title: Algebra and Discrete Mathematics

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.