Noncommutative Factorization of Variable-length Codes 1. J~ntroduction

A (variable-length) code is a free subset of a free monoid; more formally, it is the basis of some free submonoid. The theory of these codes (not to be confused with error-detecting codes) was first developed by M.P. Schiitzenberger. A major problem in this theory is to factorize codes, that is, considering the characteristic formal power series of a code (in the noncommutat ive formal power series algebra), to find a factorization of that series, or of its commutative image. Let A be some finite set and A* the free monoid generated by A. Let C be a code an g denote also by C its characteristic series, which is an element of the Z-algebra ~_%.. ~ of noncommutat ive formal power series on A. Let ~o : Z((A)) ~ Z[[A]] be the canonical homomorphism. Schiitzenberger showed that, if C is a maximal and finite code, then there exists in the algebra of commutative polynomials, Z[A], a factorization of the form