On the star-height of factor counting languages and their relationship to Rees zero-matrix semigroups

TL;DR

This paper investigates the star-height of factor counting languages and demonstrates that languages recognized by Rees zero-matrix semigroups over abelian groups have a star-height of at most one, revealing structural properties of these languages.

Contribution

It establishes that factor counting languages in specific cases have star-height at most one and links this property to languages recognized by Rees zero-matrix semigroups over abelian groups.

Findings

Factor counting languages have star-height at most one in the studied cases.

Languages recognized by Rees zero-matrix semigroups over abelian groups also have star-height at most one.

The results connect combinatorial properties of words to algebraic recognition structures.

Abstract

Given a word $w$ over a finite alphabet, we consider, in three special cases, the generalised star-height of the languages in which $w$ occurs as a contiguous subword (factor) an exact number of times and of the languages in which $w$ occurs as a contiguous subword modulo a fixed number, and prove that in each case it is at most one. We use these combinatorial results to show that any language recognised by a Rees (zero-)matrix semigroup over an abelian group is of generalised star-height at most one.