Coding Partitions of Regular Sets

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