  According to Furstenberg, an \emph{infinite parallelepiped (IP)} set in a
semigroup is a set that contains a sequence $a_1,a_2,\dotsc$, together with all
finite sums $a_{i_1}+ \dotsb + a_{i_m}$, for natural numbers $m$ and $i_1<
\dotsb <i_m$. Using a method of Galvin, Hindman proved that IP sets of natural
numbers are \emph{partition regular}: For each finite partition of an IP set,
some part must be an IP set, too. Furstenberg noted that the same proof applies
to arbitrary semigroups. However, in a semigroup with idempotents, an IP set
may be a singleton.
  A \emph{proper IP set} is one where the witnessing sequence $a_1,a_2,\dotsc$
may be chosen to be \emph{bijective} (in the naturals, every IP set is proper).
We provide a complete characterization of semigroups where the proper IP sets
are partition regular, and show that this property is equivalent to several
other fundamental notions of additive Ramsey theory.
  Using our results, we prove that if there is a monochromatic proper IP set
for each finite coloring of a semigroup, then for each finite coloring of the
semigroup there are infinitely many, pairwise disjoint, proper IP sets sharing
the same color.
