  If a finitely generated monoid M is defined by a finite number of
degree-preserving relations, then it has linear growth if and only if it can be
decomposed into a finite disjoint union of subsets (which we call "sandwiches")
of the form $a<w>b$, where $a,b,w$ are elements of $M$ and $<w>$ denotes the
monogenic semigroup generated by $w$. Moreover, the decomposition can be chosen
in such a way that the sandwiches are either singletons or "free" ones (meaning
that all elements $a w^n b$ in each sandwich are pairwise different). So, the
minimal number of free sandwiches in such a decomposition is a numerical
invariant of a homogeneous (and conjecturally, non-homogeneous) finitely
presented monoid of linear growth.
