Handbook of formal languages, vol. 1
Automata, Languages, and Machines
Automata, Languages, and Machines
Varieties Of Formal Languages
On Logical Descriptions of Regular Languages
LATIN '02 Proceedings of the 5th Latin American Symposium on Theoretical Informatics
Sur les variétés de langages et de monoídes
Proceedings of the 4th GI-Conference on Theoretical Computer Science
Shuffle on positive varieties of languages
Theoretical Computer Science
Permutation rewriting and algorithmic verification
Information and Computation
When Does Partial Commutative Closure Preserve Regularity?
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part II
Duality and Equational Theory of Regular Languages
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part II
The expressive power of the shuffle product
Information and Computation
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In this survey paper, we present known results and open questions on a proper subclass of the class of regular languages. This class, denoted by $\mathcal{W}$, is especially robust: it is closed under union, intersection, product, shuffle, left and right quotients, inverse of morphisms, length preserving morphisms and commutative closure. It can be defined as the largest positive variety of languages not containing the language (ab)*. It admits a nontrivial algebraic characterization in terms of finite ordered monoids, which implies that $\mathcal{W}$ is decidable: given a regular language, one can effectively decide whether or not it belongs to $\mathcal{W}$. We propose as a challenge to find a constructive description and a logical characterization of $\mathcal{W}$.