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

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 $\Lambda$.