On Partitions Separating Words

Partitions {Lk}k=1 of A + into m pairwise disjoint languages L1, L2, . . . , Lm such that Lk = L + k for k = 1, 2, . . . , m are considered. It is proved that such a closed partition of A+ can separate the words u1, u2, . . . , um ∈ A+ (i.e. each Lk contains exactly one word of the sequence u1, u2, . . . , um) if and only if for each pair i, j of distinct elements in {1, 2, . . . , m}, the words ui and uj do not commute. Furthermore, it is proved that the separating languages can be chosen to be regular. In case that the Parikh images of the words are linearly independent, the choice of the separating languages may be based on geometrical intuition. Formal languages; finite automata; separation by closed sets.