Regularity Conditions for Iterated Shuffle on Commutative Regular Languages

We identify a subclass of the regular commutative languages that is closed under the iterated shuffle, or shuffle closure. In particular, it is regularity-preserving on this subclass. This subclass contains the commutative group languages and, for every alphabet $\mathrm{\Sigma }$, the class ${\mathbf{\text{Com}}}^{+}({\mathrm{\Sigma }}^{\ast })$ given by the ordered variety ${\mathbf{\text{Com}}}^{+}$. Then, we state a simple characterization when the iterated shuffle on finite commutative languages gives a regular language again and state partial results for aperiodic commutative languages. We also show that the aperiodic, or star-free, commutative languages and the commutative group languages are closed under projection.