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

Commutative positive varieties of languages

We study the commutative positive varieties of languages closed under various operations: shuffle, renaming and product over one-letter alphabets

Ehrenfeucht-Fraisse Games on Omega-Terms

Fragments of first-order logic over words can often be characterized in terms of finite monoids or finite semigroups. Usually these algebraic descriptions yield decidability of the question whether a given regular language is definable in a particular fragment. An effective algebraic characterization can be obtained from identities of so-called omega-terms. In order to show that a given fragment satisfies some identity of omega-terms, one can use Ehrenfeucht-Fraisse games on word instances of the omega-terms. The resulting proofs often require a significant amount of book-keeping with respect to the constants involved. In this paper we introduce Ehrenfeucht-Fraisse games on omega-terms. To this end we assign a labeled linear order to every omega-term. Our main theorem shows that a given fragment satisfies some identity of omega-terms if and only if Duplicator has a winning strategy for the game on the resulting linear orders. This allows to avoid the book-keeping. As an application of our main result, we show that one can decide in exponential time whether all aperiodic monoids satisfy some given identity of omega-terms, thereby improving a result of McCammond (Int. J. Algebra Comput., 2001)