On shuffle products, acyclic automata and piecewise-testable languages

We show that the shuffle L \unicode{x29E2} F of a piecewise-testable language $L$ and a finite language $F$ is piecewise-testable. The proof relies on a classic but little-used automata-theoretic characterization of piecewise-testable languages. We also discuss some mild generalizations of the main result, and provide bounds on the piecewise complexity of L \unicode{x29E2} F

On the Shuffle of Star-Free Languages

Motivated by the general problem to characterize families of languages closed under shuffle, we investigate some conditions under which the shuffle of two star-free languages is star-free. Some of the special cases here approached give rise to new problems in combinatorics on words