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We investigate the state complexity of the upward and downward closure andinterior operations on commutative regular languages. Then, we systematicallystudy the state complexity of these operations and of the shuffle operation oncommutative group languages and commutative aperiodic (or star-free) languages.

State Complexity Investigations on Commutative Languages -- The Upward and Downward Closure, Commutative Aperiodic and Commutative Group Languages