A decomposition of a set $X$ of words over a $d$-letter alphabet $A=\{a_1,...,a_d\}$ is any sequence $X_1,...,X_d,Y_1,...,Y_d$ of subsets of $A^{*}$ such that the sets $X_i$, $i=1,...,d,$ are pairwise disjoint, their union is $X$, and for all $i$, $1\le i\le d$, $X_i\sim a_i{Y}_i$, where $\sim$ 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,...

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 Xi, 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...