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

Abstract

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 \(\Lambda \).