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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 acontiguous subword (factor) an exact number of times and of the languages inwhich $w$ occurs as a contiguous subword modulo a fixed number, and prove thatin each case it is at most one. We use these combinatorial results to show thatany language recognised by a Rees (zero-)matrix semigroup over an abelian groupis of generalised star-height at most one.

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