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According to Furstenberg, an \emph{infinite parallelepiped (IP)} set in asemigroup is a set that contains a sequence $a_1,a_2,\dotsc$, together with allfinite 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 naturalnumbers 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 appliesto arbitrary semigroups. However, in a semigroup with idempotents, an IP setmay 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 setsare partition regular, and show that this property is equivalent to severalother fundamental notions of additive Ramsey theory.  Using our results, we prove that if there is a monochromatic proper IP setfor each finite coloring of a semigroup, then for each finite coloring of thesemigroup there are infinitely many, pairwise disjoint, proper IP sets sharingthe same color.

Beyond the Galvin--Hindman--Furstenberg Theorem:\\ Partition regularity of IP sets in arbitrary semigroups