  Marek Kuczma asked in 1980 whether for every positive integer $n,$ there
exists a subsemigroup $M$ of a group $G,$ such that $G$ is equal to the
$n$-fold product $M\,M^{-1} M\,M^{-1} \dots\,M^{(-1)^{n-1}},$ but not to any
proper initial subproduct of this product. We answer his question
affirmatively, and prove a more general result on representing a certain sort
of relation algebra by subsets of a group. We also sketch several variants of
the latter result.
