Geometrically closed positive varieties of languages
|Článek v odborném periodiku
|Časopis / Zdroj
|Information and Computation
|Fakulta / Pracoviště MU
|Geometrical closure; Commutative closure; Variety of languages; Star-free language
|A recently introduced operation of geometrical closure on formal languages is investigated from the viewpoint of algebraic language theory. Positive varieties V containing exclusively languages with regular geometrical closure are fully characterised by inclusion of V in W, a known positive variety arising in the study of the commutative closure. It is proved that the geometrical closure of a language from the intersection of W with the variety of all star-free languages SF always falls into RLT, which is introduced as a subvariety of R, the variety of languages recognised by R-trivial monoids. All classes between RLT and W?SF are thus geometrically closed: for instance, the level 3/2 of the Straubing-Thérien hierarchy, the DA-recognisable languages, or the variety R. It is also shown that W?SF is the largest geometrically closed positive variety of star-free languages, while there is no largest geometrically closed positive variety of regular languages.