Semigroups, rings, and Markov chains

We analyze random walks on a class of semigroups called "left-regular bands ." These walks include the hyperplane chamber walks of Bidigare, Hanlon, an d Rockmore. Using methods of ring theory, we show that the transition matri ces are diagonalizable and we calculate the eigenvalues and multiplicities. The methods lead to explicit formulas for the projections onto the: eigens paces. As tramples of these semigroup walks, we construct a random walk on the maximal chains of any distributive lattice, as well as two random walks associated with any matroid. The examples include a q-analogue of the Tset lin library. The multiplicities of the eigenvalues in the matroid walks are "generalized derangement numbers," which may be of independent interest.