Finite Factors of Bernoulli Schemes and Distinguishing Labelings of Directed Graphs

A labeling of a graph is a function from the vertices of the graph to some finite set. In 1996, Albertson and Collins defined distinguishing labelings of undirected graphs. Their definition easily extends to directed graphs. Let G be a directed graph associated to the k-block presentation of a Bernoulli scheme X . We determine the automorphism group of G, and thus the distinguishing labelings of G. A labeling of G defines a finite factor of X . We define demarcating labelings and prove that demarcating labelings define finitarily Markovian finite factors of X . We use the Bell numbers to find a lower bound for the number of finitarily Markovian finite factors of a Bernoulli scheme. We show that demarcating labelings of G are distinguishing.