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Abstract: Let be an arbitrary submonoid of the free monoid , and let be a variable length code (for short a code). is weakly -complete iff any word in is a factor of some word in [J. Néraud, C. Selmi, Free monoid theory: Maximality and completeness in arbitrary submonoids, Internat. J. Algebra Comput. 13 (5) (2003) 507–516]. Given a regular submonoid , and given an arbitrary code , we are interested in the existence of a weakly -complete code that contains . Actually, in [J. Néraud, Completing a code in a regular submonoid, in: Acts of MCU’2004, Lect. Notes Comput. Sci. 3354 (2005) 281–291; J. Néraud, Completing a code in a submonoid of finite rank, Fund. Inform. 74 (2006) 549–562], by presenting a general formula, we have established that, in any case, such a code exists. In the present paper, we prove that any regular circular code may be embedded into a weakly -complete one iff the minimal automaton with behavior has a synchronizing word. As a consequence of our result an extension of the famous theorem of Schützenberger is stated for regular circular codes in the framework of regular submonoids. We study also the behaviour of the subclass of uniformly synchronous codes in connection with these questions. [Copyright &y& Elsevier]

In this paper, we discuss the different notions of local topological density for subsets of the free monoid A*. We introduce the notion of weak completeness for a set X, relatively to an arbitrary submonoid M of A*. For the so-called strongly M-thin codes, we establish that weak completeness is equivalent to maximality in M. This constitutes a new generalization of a famous result due to Schützenberger. [ABSTRACT FROM AUTHOR]