### Finite Factors of Bernoulli Schemes and Distinguishing Labelings of Directed Graphs

#### Abstract

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.