We are interested in the concept of locally complete set : A subset X of the free monoid is locally complete if a code Y in A* exists, with Y different from A, X* included in Y* and both the sets X* and Y* have the same sets of factors. Our contribution is based on the three following results: A characterization of local completeness for every thin set in terms of morphic images. A polynomial time algorithm for deciding whether a finite code is locally complete. A polynomial time algorithm for deciding whether a finite maximal code is decomposable.