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

Abstract

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.