  We provide optimal upper bounds on the growth of iterated sumsets
$hA=A+\dots+A$ for finite subsets $A$ of abelian semigroups. More precisely, we
show that the new upper bounds recently derived from Macaulay's theorem in
commutative algebra are best possible, i.e., are actually reached by suitable
subsets of suitable abelian semigroups. Our constructions, in a multiplicative
setting, are based on certain specific monomial ideals in polynomial algebras
and on their deformation into appropriate binomial ideals via Gr\"{o}bner
bases.
