We show there is a unique maximal positive variety of languages which does not contain the language (ab) *. This variety is the unique maximal positive variety satisfying the two following conditions: it is strictly included in the class of rational languages and is closed under the shuffle operation. It is also the largest proper positive variety closed under length preserving morphisms. The ordered monoids of the corresponding variety of ordered monoids are characterized as follows: for every pair (a, b) of mutually inverse elements, and for every element z of the minimal ideal of the submonoid generated by a and b, (abzab) ω ≤ ab. In particular this variety is decidable.
Los objetivos de este estudio son, primero, proponer una taxonomía de los marcadores de actitud más frecuentes utilizados en Twitter tras los comunicados del gobierno sobre el coronavirus, segundo, investigar si nos muestran que existe una aceptación del poder en España, así como un respeto a sus decisiones y, finalmente, identificar redes semánticas entre los marcadores de actitud. El corpus recopilado para el estudio está compuesto por los tuits que comentan los comunicados del Ministerio de Sanidad y Salud Pública desde el 12 al 16 de marzo de 2020. Una vez analizados los datos con las herramientas METOOL y GePhi, se observó que se desea una relación con el poder más cercana y que aumentaba la agresividad de los tuits. Finalmente, describimos las conclusiones de este estudio.
The closure of a regular language under commutation or partial commutation has been extensively studied [1,11,12,13], notably in connection with regular model checking [2,3,7] or in the study of Mazurkiewicz traces, one of the models of parallelism [14,15,16,22]. We refer the reader to the survey [10,9] or to the recent articles of Ochmański [17,18,19] for further references.In this paper, we present new advances on two problems of this area. The first problem is well-known and has a very precise statement. The second problem is more elusive, since it relies on the somewhat imprecise notion of robust class. By a robust class, we mean a class of regular languages closed under some of the usual operations on languages, such as Boolean operations, product, star, shuffle, morphisms, inverses of morphisms, residuals, etc. For instance, regular languages form a very robust class, commutative languages (languages whose syntactic monoid is commutative) also form a robust class. Finally, group languages (languages whose syntactic monoid is a finite group) form a semi-robust class: they are closed under Boolean operation, residuals and inverses of morphisms, but not under product, shuffle, morphisms or star.Here are the two problems: The classes considered in this paper are all closed under polynomial operations. Recall that, given a class L of regular languages, the polynomial languages of L are the finite unions of languages of the form L 0 a 1 L 1 · · · a k L k where a 1 , . . . , a k are letters and L 0 , . . . , L k are languages of L. Taking the polynomial closure usually increase robustness. For instance, the class Pol(G) of polynomials of group languages is closed under union, intersection, quotients, product, shuffle and inverses of morphisms. Let I be a partial commutation and let D be its complement in A × A. Our main results on Problems 1 and 2 can be summarized as follows:
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Abstract. 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 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 W is decidable: given a regular language, one can effectively decide whether or not it belongs to W. We propose as a challenge to find a constructive description and a logical characterization of W.Warning. In this paper, square brackets are used as a substitute to "respectively" to gather several definitions [properties] into a single one.The search for robust classes of regular languages is an old problem of automata theory, which occurs in particular in the study of regular model checking [3]. In this survey paper, we present known results and open questions on a proper subclass of the class of regular languages, introduced a few years ago by the authors in connection with the study of the shuffle product [6,7]. This class, denoted by 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. Furthermore, this class is decidable: there is an algorithm to decide whether a given regular language belongs to W or not. As such, it might offer an appropriate framework for modeling certain problems arising in the verification of concurrent systems.The class W is also interesting on its own and appears in the study of three operations on languages: length preserving morphisms, inverse of substitutions and shuffle product. More specifically, W is the largest proper positive variety of languages closed under one of these operations. It is also the largest positive variety of languages not containing the language (ab) * .⋆ The authors acknowledge support from the AutoMathA programme of the European Science Foundation.
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