Given a hyperplane arrangement in Rn and the corresponding cell decomposition, Bidigaire defined a semigroup structure on the set of faces of the cell decomposition and used this semigroup to calculate the eigenvalues, with multiplicities, for a random walk on the chambers. The particular case of the braid arrangement leads to a famous Markov chain from computer science, called the Tsetlin library. These results were further extended by Ken Brown and Persi Diaconis who showed the diagonalizability of the transition matrix and calculated the stationary distribution.