For every Abelian group $(S, +)$ there exist Boolean---i.e.,
compact, 0-dimensional---topological spaces $X_{s},~s\in S$, such
that $s+t=u$ if and only if $X_{u}$ is homeomorphic to the disjoint
union of $X_{s}$ and $X_{t}$. The method of the proof of this
theorem is topological, utilizing mostly properties of \v{C}ech-Stone
compactifications of various spaces. A corollary, obtained
from well-known dualities, is the representability of Abelian
groups (in an analogous sense) by products of rings, lattices,
Boolean algebras, Banach spaces or Banach algebras.
