Vol. 61, No. 1, 1975

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Sums of Boolean spaces represent every group

Jiří Adámek, V. Koubek and Vĕra Trnkov\'{a}

Vol. 61 (1975), No. 1, 1–6
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.
Mathematical Subject Classification 2000
Primary: 06A40, 06A40
Secondary: 20K99, 08A10
Received: 6 September 1974
Revised: 14 February 1975
Published: 1 November 1975
Jiří Adámek
V. Koubek
Vĕra Trnkov\'{a}