The Degree of a Finite Set of Words

We generalize the notions of the degree and composition from uniquely decipherable codes to arbitrary finite sets of words. We prove that if X = Y ◦ Z is a composition of finite sets of words with Y complete, then d ( X ) ≤ d ( Y ) · d ( Z ), where d ( T ) is the degree of T . We also show that a finite set is synchronizing if and only if its degree equals one. This is done by considering, for an arbitrary finite set X of words, the transition monoid of an automaton recognizing X ∗ with multiplicities. We prove a number of results for such monoids, which generalize corresponding results for unambiguous monoids of relations.