Some results on the generalized star-height problem

Abstract

We prove some results related to the generalized star-height problem. In this problem, as opposed to the restricted star-height problem, complementation is considered as a basic operator. We first show that the class of languages of star-height ≤ n is closed under certain operations (left and right quotients, inverse alphabetic morphisms, injective star-free substitutions). It is known that languages recognized by a commutative group are of star-height 1. We extend this result to nilpotent groups of class 2 and to the groups that divide a semidirect product of a commutative group by ( Z /2 Z ) n . In the same direction, we show that one of the languages that were conjectured to be of star-height 2 during the past ten years is in fact of star-height 1. Next we show that if a rational language L is recognized by a monoid of the variety generated by wreath products of the form M ∘ ( G ∘ N ), where M and N are aperiodic monoids, and G is a commutative group, the L is of star-height ≤ 1. Finally we show that every rational language is the inverse image, under some morphism between free monoids, of a language of (resticted) star-height 1.