A Note on the Commutative Closure of Star-Free Languages

We show that the commutative closure of a star-free language is either star-free or not regular anymore. Actually, this property is shown to hold exactly for the closure with respect to a partial commutation corresponding to a transitive dependence relation. Moreover, the question whether the closure of a star-free language remains star-free is decidable precisely for transitive partial commutation relations. It is well-known that the commutative closure of regular languages need not be regular. A typical example is the star-free(!) language (ab) , the commutative closure of which consists of all words over fa; bg with equal number of a and b. Moreover, as soon as the closure of languages under partial commutation is considered the question whether the closure of a given regular language remains regular or not is in general undecidable 9]. On the other hand, the question whether the closure of the star of a closed regular language remains regular or not is still an open question for partial commutations. Recently, some progress has been achieved towards a solution for this problem 6, 8] (for a survey and further references see also 2, Chapter 6]). The aim of this paper is to clarify another aspect concerning the still mysterious behaviour of recognizable languages with respect to closure operations, namely the relationship between star-freeness and closure under commutation. We show that surprisingly there are no star-free languages, where the commu-tative closure is regular, but not star-free. In fact, we can show a more general result concerning the closure of star-free languages under partial commutations.