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

• 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: emiperfect ring; tiled order; quiver; partially ordered set; index of semiperfect ring; Gorenstein tiled order; finite Markov chain; 16P40; 16G10
• Online Access: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/3
• Journal Title: Algebra and Discrete Mathematics

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 \(\Lambda \).