  It is shown that a flat subgroup, $H$, of the totally disconnected, locally
compact group $G$ decomposes into a finite number of subsemigroups on which the
scale function is multiplicative. The image, $P$, of a multiplicative semigroup
in the quotient, $H/H(1)$, of $H$ by its uniscalar subgroup has a unique
minimal generating set which determines a natural Cayley graph structure on
$P$. For each compact, open subgroup $U$ of $G$, a graph is defined and it is
shown that if $P$ is multiplicative over $U$ then this graph is a regular,
rooted, strongly simple $P$-graph. This extends to higher rank the result of R.
M\"oller that $U$ is tidy for $x$ if and only if a certain graph is a regular,
rooted tree.
