Geometrical Closure of Binary $$V_{3/2}$$ Languages

Geometrically Closed Positive Varieties of Star-Free Languages

“…We denote by [w] In the previous paragraph we consider the mapping Ψ : Σ * → N k , where N is the set of all non-negative integers. Following the ideas of [8], we introduce some technical notations concerning N k , whose elements are called vectors. We denote by 0 the null vector of N k .…”

“…A geometrical closure is an operation on formal languages introduced recently by Dubernard, Guaiana, and Mignot [8]. It is defined as follows: Take any language L over some k-letter alphabet and consider the set called the figure of L in [8], which consists of all elements of N k corresponding to Parikh vectors of prefixes of words from L. The geometrical closure of L is the language γ(L) of all words w such that the Parikh vectors of all the prefixes of w lie in the figure of L. This closure operator was inspired by the previous works of Blanpain, Champarnaud, and Dubernard [4] and Béal et al [3], in which geometrical languages are studied -using the terminology from later paper [8], these can be described as languages whose prefix closure is equal to their geometrical closure. Note that this terminology was motivated by the fact that a geometrical language is completely determined by its (geometrical) figure.…”

“…Note that this terminology was motivated by the fact that a geometrical language is completely determined by its (geometrical) figure. In the particular case of binary alphabets, these (geometrical) figures were illustrated by plane diagrams in [8].…”

“…The class of all regular languages can be easily observed not to be geometrically closed -that is, one can find a regular language such that its geometrical closure is not regular [8] (see also the end of Sect. 2).…”

“…One possible research aim could be to characterise regular languages L for which γ(L) is regular, or to describe some robust classes of languages with this property. Another problem posed in [8] is to find some subclasses of regular languages that are geometrically closed. As we explain in Sect.…”