Coding partitions of regular sets

A coding partition of a set of words partitions this set into classes such that whenevera sequence, of minimal length, has two distinct factorizations, the words of these factorizationsbelong to the same class. The canonical coding partition is the finest codingpartition that partitions the set of words in at most one unambiguous class and otherclasses that localize the ambiguities in the factorizations of finite sequences.We prove that the canonical coding partition of a regular set contains a finite numberof regular classes and we give an algorithm for computing this partition. From this wederive a canonical decomposition of a regular monoid into a free product of finitely manyregular monoids.