Editing Free monoid (section)

==Free hull==
The intersection of free submonoids of a free monoid ''A''<sup>&lowast;</sup> is again free.<ref name=Lot6>{{harvtxt|Lothaire|1997|p=6}}</ref><ref name=LotII204>{{harvtxt|Lothaire|2011|p=204}}</ref>  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 and called the '''free hull''' of ''S''. A basis for this intersection is a code.

The '''defect theorem'''<ref name=Lot6/><ref name=LotII204/><ref name=BPR66>{{harvtxt|Berstel|Perrin|Reutenauer|2010|p=66}}</ref> states that if ''X'' is finite and ''C'' is the basis of the free hull of ''X'', then either ''X'' is a code and ''C'' = ''X'', or

:|''C''| ≤ |''X''| − 1 .