Free Hull
The intersection of free submonoids of a free monoid A∗ is again free. If S is a subset of a free monoid A* then the intersection of all free submonoids of A* containing S is well-defined, since A* itself is free, and contains S; it is a free monoid. A basis for this intersection is the free hull of S.
The defect theorem states that if X is finite and C is the free hull of X, then either X is a code and C = X, or
- |C| ≤ |X| − 1 .
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