| The closure of a regular language under commutation or partial commutation has been extensively studied. In this paper, we present new advances on two problems of this area. Problem 1. When is the closure of a regular language under [partial] commutation still regular? Problem 2. Are there any robust class of languages closed under [partial] commutation? Let A be a finite alphabet, I a partial commutation on A and D be its complement in A x A. Our main results on Problems 1 and 2 can be summarized as follows: The class Pol G (polynomials of group languages) is closed under commutation. If D is transitive, it is also closed under I-commutation. Under some simple conditions on the graph of I, the closure of a language of Pol G under I-commutation is regular. There is a very robust class of languages W which is closed under commutation. This class, which contains Pol G, is closed under intersection, union, shuffle, concatenation, residual, length preserving morphisms and inverses of morphisms. Further, it is decidable and can be defined as the largest positive variety of languages not containing (ab)*. If I is transitive, the closure of a language of W under I-commutation is regular. The proofs are nontrivial and combine several advanced techniques, including combinatorial Ramsey type arguments, algebraic properties of the syntactic monoid, finiteness conditions on semigroups and properties of insertion systems. |
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