Theoretical Computer Science 12 (1980) 325-332 @ North-Holland Publishing Company zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE NOTE MINIMAL COMPLETE SETS OF WORDS ” Jean Marie BOfi zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Universite des Sciences et Techniques du Languedoc, M ontpellier, France Aldo de LUCA and Antonio REST1170 Laboratorio di Cibemetica de1 CNR, Arco Felice, Napoli, Italy Communicated by M. Nivat Received October 1979 Abstract. is givea. A characterization of minimal complete sets of words of a free monoid which are codes, zyxwvutsrqpon 1. Introduction A subset X of a free monoid zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC A* is complete if X*, the submonoid generated by zyxwvutsrqponm X, is dense, i.e. X* meets all two-sided ideals of A*. IF. other words a set X is complete if any word of A* is a factor of some word of X*a Complete sets play an interesting role in some problems of language theory. In particular the notion of ‘complete set’ is in a sort of dual correspondence with the notion of ‘code’. Indeed a basic result of M.P. Schiitzenberger states that any maximal code is a complete set and any nondense complete code is a maximal code. From this result one derives that any nondense maximal code is a minimal complete set, i.e. a complete set that does not contain complete proper subsets. We recall that maximal codes are utilized in information theory in order to obtain optimal rates of transmission. In this paper we pose the question as to whether the converse of the previous assertion holds, i.e. whether a minimal complete set is a maximal code. A negative answer to this question and a characterization of minimal complete sets which are codes are given. More precisely our m in result states that a minimal comp!eie set is a maximal code if and only if all its proper subsets are codes. 2. Complete sets an Let A be a finite alphabet and A* the free monoid generated by it. As usua! the elements of A are catled letters and the elements of A* words. The identity element 325