  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.
