A decomposition of a set X of words over a d-letter alphabet
A = {a1,...,ad} is any sequence X1,...,Xd,Y1,...,Yd
of subsets of A* such that the sets X
i
, i = 1,...,d, are pairwise disjoint,
their union is X, and for all i, 1 ≤ i ≤ d, Xi ~ aiYi
, where ~
denotes the commutative equivalence relation. We introduce some suitable decompositions
that we call good, admissible, and normal. A normal decomposition is admissible and
an admissible decomposition is good. We prove that a set is commutatively
prefix if and only if it has a normal decomposition. In particular, we consider
decompositions of Bernoulli sets and codes. We prove that there exist Bernoulli sets
which have no good decomposition. Moreover, we show that the classical conjecture of
commutative equivalence of finite maximal codes to prefix ones is equivalent to the statement
that any finite and maximal code has an admissible decomposition.